The Home Office has rejected a family’s plea to allow the grandparents of a five-year-old girl to come to Britain to attend her funeral despite a personal promise hours before by David Cameron to intervene in the case.

The Zimbabwean grandparents and aunt of Andrea Gada, who died after being hit by a car just before Christmas in Eastbourne, Sussex, have been refused temporary travel visas to attend her funeral.

The immigration minister, James Brokenshire, turned down a public offer by the family’s MP to act as a personal guarantor that Andrea’s relatives would return home to Zimbabwe after her funeral.

The Bishop of Chichester has condemned Brokenshire’s decision, telling him: “This cannot be right. It offends at the most elementary level of human compassion.”

Andrea’s father, Wellington, said: “Ever since our beloved daughter was tragically killed we have tried in vain to get my wife’s father, mother and sister who live in Zimbabwe to attend her funeral. This would give our two remaining children, and our ourselves, much needed comfort at this grieving moment.”

The Eastbourne community, including Andrea’s school, Shinewater primary, have rallied around her refugee parents, Wellington and Charity Gada, to raise more than £5,000 to enable her relatives to travel. Andrea’s funeral has had to be postponed until the situation is resolved.

Cameron promised he would review the case after the family’s MP, Stephen Lloyd, raised it at prime minister’s question time last Wednesday. The Liberal Democrat MP for Eastbourne told Cameron the entry visas had been refused because the Home Office believed the relatives would abscond, and offered to personally guarantee that they would return to Zimbabwe.

Cameron promised to look into the case, saying it was heartbreaking when children were killed in accidents: “I will certainly look at the case – I was just discussing it with the home secretary – and make sure that the Home Office has a careful look to see what can be done.”

But a day later, Brokenshire wrote to Lloyd, telling him: “I am sure you will understand that I am often asked to exercise discretion in individual cases. I am also frequently offered assurances or guarantees, despite the tragic circumstances involved.”

The minister said he had reviewed the case, “taking into account the tragic circumstances”, but his decision had to be based “on the full facts of the case”. As the family members had not provided “evidence to demonstrate that they meet the requirements of the immigration rules” he could not overturn the refusal.

Lloyd said the visas had been refused for three reasons: they had not travelled out of Zimbabawe before, they could not demonstrate a regular income, and therefore there was a danger they would abscond while in Britain. But he said it was not surprising that the three – a street trader, a hairdresser and a driver – could not demonstrate a regular income in Zimbabwe. The grandparents had also previously travelled to South Africa and returned home.

He said he was disappointed that the Home Office had sent their second rejection letter within 24 hours of Cameron’s promise to look at the case: “What the Home Office is really saying is that the grandparents and aunt of Andrea can’t be trusted with a temporary travel visa to attend the funeral and return to Zimbabwe afterwards because they are ‘too poor’. That view is simply wrong on every level. I am asking David Cameron to do the right thing for Andrea and her family.”

Andrea’s father said he was devastated when he got the Brokenshire letter on Friday. He intends to petition the prime minister, telling him that he is even prepared to see his relatives electronically tagged if it would reassure the Home Office they would not abscond.

The Bishop of Chichester, the Rt Rev Dr Martin Warner, said in his letter of support that the Gadas were not rich or influential people and that the family were Seventh-Day Adventists for whom the funeral rite was more than a ceremony: “What are the grounds for refusal of this consolation in grief?” he asked.