Inspector David Gilbert, spokesman for the investigating team, said he had 'no remit' to discuss the concerns. He suggested any answers could compromise evidence to be presented in court.
A 22-year-old woman has been charged with abducting Abbie. She will appear at Nottingham magistrates' court today. A man and woman have been released without charge.
Insp Gilbert stuck to an earlier police statement which admitted that officers had previously visited the detached house in Brendon Drive from which Abbie was rescued, two weeks after she was kidnapped. But he refused to reveal how many times. Neighbours say that police visited the house on two previous occasions.
Yesterday, three local women - a neighbour, midwife and childminder - said they had to push the police to make a return visit to the house. Although police were satisfied by explanations offered by the occupants, the women remained convinced Abbie was there. They say they made their own investigations and presented police with enough information to return to the house.
The claims were made in the Sunday Mirror, which offered pounds 25,000 for information leading to Abbie's safe return.
On Saturday, Karen Humphries, 32, and her husband Roger, 33, were full of praise for the police. Officers are annoyed by the doubts now raised about their handling of the inquiry and believe the difficulties of the rare and delicate case are not being appreciated. They insist Abbie's safe return is proof that their strategy was the right one. During the inquiry they relied heavily on personal appeals to the abductors and her relatives and friends and continually emphasised the value of public vigilance.
The Humphries, who also have a three-year-old son, Charlie, spent yesterday with their family and friends. On Saturday night they celebrated Abbie's return with a barbecue.
'They're still walking on the ceiling,' said a police spokesman. 'Now they just want to be left to lead a normal life.'
Several tabloid newspapers are now competing for the Humphries' story of the 15-day separation from their new-born daughter.
Abbie was taken from the maternity wing of the Queen's Medical Centre, Nottingham, on 1 July by a bogus nurse, who persuaded her father to hand her over for a 'routine hearing test'.
Early on Saturday morning, police raided a detached house a mile from the medical centre and recovered the baby. She was safe and well.
At a press conference on Saturday, Mrs Humphries said a special thank-you to the woman who had eventually come forward with the information that led to Abbie. It now appears that the tip-off was perhaps one of a series of approaches made by the same woman to the police.
A charity said last night that it was counselling a woman who wants to abduct and kill a baby.
The Portia Trust, which expects there to be about 20 'copycat' abduction attempts in the wake of the Abbie case, was contacted on Friday by the distraught woman, who confessed to a psychopathic illness which made her want to harm a child.
A spokesman said: 'The woman who wants to kill a child is the first we have had experience of, but we will do all we can to help her. We certainly will not reveal where she lives, or contact the police, since we guarantee to keep confidential all contacts.'Reuse content