Only 1 in 172 fathers in the UK are taking APL, the Trade Union Congress said, citing the study.
The study for 2011/12 found just 1,650 (0.6 per cent) out of 285,000 dads took the leave at the statutory rate of 136 pounds a week.
Under APL a father is able to take up to 26 weeks leave and 19 weeks pay when his partner has ended her maternity leave, no earlier than 20 weeks after the birth of the baby.
The system which allows men to take up to 26 weeks additional paternity leave - and receive additional statutory paternity pay - was only introduced in April 2011 and the TUC figures are based on its first year.
More From This Section
New fathers had already been entitled - and continue to be entitled - to two weeks ordinary paternity leave. They can only claim the extra weeks in their child's first year if the mother returns to work before taking her full maternity leave entitlement.
"Poor levels of financial support are preventing new dads from taking extra time off and are particularly affecting low -paid fathers who simply cannot afford to take leave.
"Extending paternity pay from two to six weeks and paying a better statutory rate would make a massive difference, as has been shown in other countries."
The TUC said in contrast the first two weeks of paternity leave was taken by nine out of 10 fathers. But the difference is that although the statutory rate is the same - 136 pounds a week - many employers typically choose to top this up to full pay throughout that short period of leave.
"This is why we are introducing a system of shared parental leave from April 2015 so that fathers can take more leave if they want to in the early days of a child's life.
"We want to challenge the myth that it is the mother's role to stay at home and care for children," the BBC quoted the spokesman as saying.