I can think of a couple of terms, but each would have the renter, not the landlord as the agent.
This is mostly British usage. From the OED
And additional or extra amount or payment that restores something to
the level that is required.
“They will miss out on hundreds of pounds worth of pension top-ups”
Informal, this is from the urban dictionary, not an actual dictionary.
To correct any skewed perception or actuality
Uses: true-up, truing-up, trued-up, true up, truing up, trued up A
large company buys X number licenses of software program for use by
the employees. As the company grows the software program gets copied
and used, therefore cheating the software makers out of money. Yearly,
the company needs to "true-up" the copies being used and pay the
software makers for what they are actually using.
This term is more commonly used by car and bicycle mechanics, but it can be used for money too.
Ps: I don’t know this for sure, but my guess is that this informal use (in quotes) in the context of money comes from the actual use of the term “true up” in accounting, with the meaning “to make an entry to reconcile a calculation”.
Since you need the landlord to be the one making the decision, could you perhaps frame your sentence as “The landlord decided to make him top up the deposit.”?