Families are faced with moving to cities as far afield as Grimsby as the new benefit cap hits, along with the high cost of renting in the capital

This month Nikola Gardner, 41, will be getting ready for another move with her children – her fifth in four years – as changes to her benefit allowances edge her further from the patch of central London where she was born. She is living now on a quiet street in the outer London borough of Enfield, but she knows that soon there will be no place left for her in the capital. She fears the next move, as a result of the new benefit cap, could see her dispatched as far as Grimsby.

As concern rises about soaring property prices in the capital, and the increasing unaffordability of housing for those hoping to buy, there also are severe consequences for those at the bottom end of the London property chain, far removed from even contemplating buying somewhere to live. Rising house prices have led to soaring rents, pushing up the government's housing benefit bill, and triggering in turn radical cuts to the amount of support available for those who need it.

Gardner moved from Islington in February 2012 to Enfield, 12 miles from her family, after being evicted from her home when a cut to the amount she was entitled to made the flat she was renting unaffordable.

But the newly introduced overall benefit cap of £500 a week, now means the home she settled in with her two sons (aged four and eight) has now in turn become unaffordable, so she is set to be evicted for a second time.

Her situation is not unique. Her friend, Clarissa Pilgrim, 36, was also rehoused in Enfield by Islington council in September 2012, when her flat became too expensive. She was helped to find a private flat to rent, paying for it with housing benefit. Less than 18 months after moving, the introduction of the benefit cap means she is no longer able to pay her rent, and she too faces eviction for a second time.

The homelessness charity Crisis recently revealed that the main cause of homelessness in England is now people's private tenancies coming to an end, with many unable to find somewhere else to live because benefit changes have made it impossible for tenants to keep up with soaring rental costs.

Karen Buck, Labour MP for Westminster North, says it is not uncommon to see households making "very frequent, involuntary moves" as a result of a series of benefit changes. Research by the New Policy Institute shows changes to benefits mean that almost all of London is now unaffordable to a family with two children who are supported by housing benefits.

Gardner knows that discussion of benefits has become a vitriolic theme, so she is at pains to set out why she is currently, and she hopes just temporarily, being supported by housing benefit.

She was working until five years ago, as a deputy manager with National Car Parks in Gatwick. She had always worked, but relied on her mother to help her with childcare while she was at work, picking up her children from nursery, particularly after she split up from their father. When her mother was diagnosed with cancer and died, she was devastated, but also found herself unable to afford the childcare to allow her back to work.

She was living in a three-bedroom flat in Islington, which cost £550 a week, a rent which was initially met by housing benefit in full. "I was gobsmacked when I found out how much it cost. It was not a nice place. It was not a luxury place," she says. In March 2012, a new cap reduced the amount of housing benefit payable to around £350 a week for a three-bedroom property – so she would have had to make up the extra £200 a week. Since she was an unemployed single mother with two young sons, there was no way she could find that money, so she had to move.

Council officials told her that she would be rehoused in Enfield, where rents were cheaper. She was moved into a temporary flat in the borough, and for several months got up at 5.30 every morning to travel with her son to his old school (travelling back and forth by train was too expensive, so they went by bus, which was very time-consuming). When she was offered something permanent there, she accepted it, and switched him to a local school, reassured by the council that she would be able to stay in the area. Once she acclimatised to the move, she was surprised to find she quite liked the quiet area of Bush Hill Park, despite her sense of isolation.

However, only a year later, new changes to the benefit system mean that the £300 a week rent on the house where she now lives pushes her above the £500 ceiling, leaving her with a shortfall of £80 a month, which she is unable to find out of her other benefit payments. She has been granted a temporary discretionary housing payment to meet the shortfall, but when this runs out, she will have to move, further out again. Birmingham, Manchester and Grimsby have all been mentioned as possible options, she says.

"It was a nightmare being pushed up here, but now we're settled, we don't want to move again. I don't know if the children will cope with moving again."

But she knows she will not be able to find the extra money required. She realises that because of the culture of hostility to people supported by benefits, the disruption faced by her family will elicit very little sympathy.

"It has got worse. When people hear you are on benefits, they say: 'it's not right that people should be on benefits and living the life'. But what life am I living? I have two sets of clothes – leggings and a top. I don't feel like I'm existing. I'm someone who wakes up, takes the children to school, cleans, picks them up, cooks for them, sleeps. I don't see anyone, or go anywhere."

She wants to go into full-time work when both her children are at school full time; her youngest, who is four, currently gets just 15 hours a week of state-funded childcare.

Gardner thinks people don't understand that she did not choose to live in the most expensive part of the UK and possibly Europe. "We were born here, we were raised here, brought up here, it's all we know. I've already moved away from friends to come here. How would I afford to see family if we leave London?"

Pilgrim agrees. "Rent is stupid in London. There is nowhere affordable to move into. We didn't choose to move into a place that cost £550 a week," she says. "It's not fair to move children away from what they know."

Pilgrim worked as a parent support adviser before she was made redundant in a wave of government cuts. Later, she worked as a teaching assistant in a school and a youth worker in the evening. She stopped working after her second child was born, and wants to return to work when her daughter goes to school full-time. At the moment she has only three hours of nursery a day. "What can you do in three hours?"

She was living in Islington in a flat that cost £425 a week, when the first cap was introduced, and was moved to Enfield in 2010 to take a house priced at £310 a week. She was also assured that she would not be moved on again, once new reforms came in. But now the £26,000 benefit cap has been introduced she is about £100 a month short. She is using savings from when she was working to help meet the gap between rent and benefits. She has also cut down on the heating. "It goes on for 15 minutes at 5.45 am, for 15 minutes at 10am, for 15 minutes when they come home from school at 3 and for half an hour at six o'clock."

She is juggling disconnection notices from utility providers. "I always have a threatening letter. I feel trapped. I have no support network here, even if I wanted to go back to work."

For the moment, she is coping with the shortfall but the bills are mounting up. "It will take one thing for it to crumble. I'm just teetering on the edge," she says. She has looked at rents in Chesham, 29 miles away out of London, which look more affordable but is dreading moving her family again, so soon after the last move.

When changes to housing benefit allowances were introduced, the government said that they would help to push down private rents. However, Lord Freud, work and pensions minister, conceded last month that the policy had not led to a reduction in private sector rents.

Citizens Advice chief executive, Gillian Guy, says: "The eye-watering cost of housing can push some households over the edge. We've seen rent arrears problems reported to our bureaux go up in every region of England since last year, with a 13% increase across the country for social housing tenants.

"Ministers' efforts at getting the cost of housing benefit down far too often punish a small group of people without getting to the root of the housing problem in our country, which is the dire lack of affordable homes. Pushing up costs for some families without tackling supply is an upside-down approach."

An Islington council spokesman says: "We place people in temporary accommodation in Islington when we can, but because of a shortage of housing this isn't always possible."