My tenancy agreement says I have to pay for a professional clean when I move out - is that allowed?

No, a clause like that isn't enforceable. Since the Tenant Fees Act 2019 a landlord in England can't make you pay for a professional clean as a condition of the tenancy, even when it's written into the agreement. What you actually owe is simpler: hand the property back as clean as it was when you moved in, allowing for fair wear and tear. That's the real test, and it's why the check-in inventory matters far more than any cleaning clause.
The first thing I'd check is whether that inventory exists and how detailed it is. If there's no inventory, a landlord has very little to stand on when they try to deduct for cleaning. If there is one, work through it room by room. The oven, the extractor, inside the kitchen cupboards, limescale on taps and shower screens, skirting boards and window tracks are the spots that catch people out and trigger deposit disputes.
You don't have to hire anyone, and plenty of tenants get the full deposit back doing it themselves over a weekend. But if you're short on time, or the oven's had two years of roast dinners baked onto it, a professional clean simply takes the argument away. Expect somewhere around £120 to £300 for a typical flat or small house, depending on size and the state of that oven. Keep the receipt and before-and-after photos either way, because that's what settles it if the deposit scheme gets involved.
If you'd rather hand it over, we can match you with vetted end-of-tenancy cleaners near you so you can compare quotes. To sanity-check any price first, see our end of tenancy cleaning cost guide.