If so, how much is the fee? Tenant Fees Act 2019 provides a special procedure and form TFA1 to recover banned fees, but must one pay a fee to file such an application?
Yes, it's £100.
See the schedule to the Civil Proceedings, First-tier Tribunal, Upper Tribunal and Employment Tribunals Fees (Amendment) Order 2016 which substitutes Schedule 1 of the First-tier Tribunal (Property Chamber) Fees Order 2013 to include:
Column 1 Number and description of fee
1.1 Where no other fee is specified, on filing an application to commence proceedings in any leasehold case or on filing an appeal or an application to commence proceedings in a residential property case.
Column 2 Amount of fee