Sounds like how it works in my locale.

You used the agent. Did you think the agent worked for free? Did you not ask what it would cost in advance? Of course you pay for it.

If you contracted with the agent to find you a tenant then you pay. and 1 month’s rent is customary, In NYC however there is the illegal institution of “Key money” which is unrecorded , non-refundable cash payment from the new tenant paid to either the Real Estate person as a finder’s fee, or to the landlord for choosing them as the new tenant, and sometimes paid to an old tenant to get them to vacate. Next time you should try and collect some of this gravy.
If the tenant does not stay the full 1st year of the tenancy (lease term) the real estate agent is usually required to return a prorated portion of their fee, but their nature being what it is , good luck trying to collect this without having to go to court to get it.

If you hired the agent then you have to pay the agent.

sorry. yes, you should have worked that out the the agent first.
we looked as using brokers recently for renting in nyc.
normally
it is first and last to owner
and either one month rent to broker
or 10 percent of annual rent.
too late now to fix it, but if you work with an agent again–figure it out in writing.
you cant ask of anything from the tenant now.

How in the world did you get to where you are in your business? You should know when you lease direct to a tenant YOU are your own agent. When you rent through an agent you must pay the agent. Perhaps you should go back and gain a little more education in your line of work.

The industry standard in Manhattan is for the real estate agent to charge the lessee 12-15% of the annual rent. The lessor (you) typically requires 1-2 months of security deposit, but never collects any fee for renting the apartment. One alternative is to go direct to lessee’s and then just charge a premium on the rent, but game it by not charging the full 12-15% equivalent uplift.

If you recruited the real estate agent, you should pay.