See this government document, seems pretty clear to me.
Sounds like they’re trying to see what they can get away with, but the below is clear on the payments you can make.
The only payments in connection with a tenancy that you can be asked to
• the rent
• a refundable tenancy deposit capped at no more than five weeks’ rent where
your total annual rent is less than £50,000, or six weeks’ rent where your total
annual rent is £50,000 or above
• a refundable holding deposit (to reserve a property) capped at no more than
one week’s rent
• payments to change the tenancy when requested by the tenant, capped at
£50, or reasonable costs incurred if higher
• payments associated with early termination of the tenancy, when requested
by the tenant
• payments in respect of utilities, communication services, TV licence and
council tax; and
• A default fee for late payment of rent and replacement of a lost key/security
device giving access to the housing, where required under a tenancy
edit: I suppose they could say that this has nothing to do with the tenancy and is a fee they want to charge you to do business with the letting agent in general? If so, that still sounds pretty shady, but it could be a work around. In my opinion, it’s still technically in connection with a tenancy.