The government is “reviewing” the surcharge that some migrant doctors and nurses working in the NHS must pay to access the health service, the home secretary Priti Patel has said.

In the daily press conference at Downing Street on Saturday, after the UK became the fifth country to pass 20,000 total deaths from coronavirus – behind the US, Italy, Spain and France – Patel reiterated the government’s stay-at-home message and rejected suggestions that allowing non-essential work to continue sent mixed messages.

The government has been criticised for planning to increase the immigration health surcharge for migrants from outside the European Economic Area, including nurses and doctors, from £400 to £625 a year each from October. The fees were introduced in 2015.

The move was described as a perverse deterrent to workers the government needed to attract, and there have been reports of nurses facing fees of thousands of pounds a year in both visa charges and the surcharge so they and their families can access the health service and remain in the UK.

Asked if this was the right time to abolish it for NHS workers, Patel said: “You’ll be aware of many changes we have already made around the immigration status, the visa status for NHS workers, extending their visas already if they were coming up for expiry.

“We have a range of measures that are, like most things in government, under review, and we are looking at everything including visas, surcharge … We are looking at everything we can do to continue to support everyone on the frontline in the NHS.”

At the end of March, the Home Office said that NHS workers – and their family members – would automatically have their visas extended free of charge for one year.

Speaking at the daily briefing for the first time, Lynne Owens, the director general of the National Crime Agency (NCA), also warned that “amoral, corrupt and exploitative” criminals were seeking to take advantage of the crisis.

More than 2,000 online scams relating to coronavirus have been taken down including fake shops selling bogus testing kits and PPE, she said.

Patel sought to warn criminals seeking to exploit the current situation that they would be pursued. She said the NCA had flagged up 1,300 possible child sexual offences to police.

In her closing remarks, the home secretary said: “Any criminal seeking to exploit this virus for their own gain, our outstanding police and law enforcement agencies are absolutely on to you.”

The press conference came amid apparent concern in government about waning respect for physical distancing measures.

NHS England’s medical director, Stephen Powis, exhibited a slide showing a slight increase in the number of journeys taken and warned people to avoid travel and to physically distance – with the measures now in place until at least 7 May – during the UK’s fifth weekend under lockdown.

The Home Office said: “We are immensely grateful for the work NHS doctors, nurses and paramedics are doing to tackle coronavirus, and this is why we applied a one-year visa extension to all those whose visas are due to expire before 1 October 2020 – this is completely free of charge and includes an exemption from the immigration health surcharge.

“We will continue to look at whether we can provide further assistance during the fight against this virus.”

• This article was amended on 29 April 2020. An earlier version said the UK had become the fifth country to pass 20,000 deaths in hospital from coronavirus. This should have referred to all deaths, not just those in hospital. This has been corrected.