Do we have to pay rent for a week the letting agency chose to leave the property empty?

I’m trying to work out whether my letting agency is acting legally here or if they’re trying to charge us for things they can’t actually charge for.

We’re ending our tenancy early and originally gave the required two months’ notice. Our tenancy would then end on 18 July.

The director of the letting agency then contacted us and asked whether we’d be willing to leave earlier because they didn’t have any available properties for June and had a prospective tenant who wanted to move in. This actually suited us because we were moving into a new place on 1 June anyway. Based on that conversation, we agreed to leave earlier as it seemed beneficial for everyone involved. (This was in person so nothing is written).

We said yes priorities a move in for June as that’s great for us. I mean our rent before was £2400 & it’s now being rented for £2750. So they’re making their money there.

Now the agency is saying that:

We must give up possession of the property on the day our tenancy ends but -
We must continue paying rent until 24 June, when the new tenant moves in.
We must also effectively pay for an additional week where the property will be vacant so they can carry out works before the new tenant moves in.

The issue is that we never agreed to pay for a vacant week. None of these dates were discussed with us beforehand, and there is nothing in writing saying we’d be responsible for a period where the property is empty purely because the agency wants time to do works.

Our tenancy agreement does state that if we end the tenancy early, we are liable for rent until a replacement tenant is found. However, a replacement tenant has been found. The agency chose the move-in date and chose to leave a week’s gap between tenancies without informing us prior that we’d be liable for that. We didn’t request this.

To make things even more confusing, they were the ones encouraging the earlier move-out and calling us about this new tenant, saying the tenant was keen to move in and asking whether we were still happy to leave early.

They’re also trying to charge us for a professional clean (£350) and are saying we need to steam-clean carpets that were brand new when we moved in. They’re effectively saying the property must be returned in exactly the same condition as when we moved in, including the carpets, which doesn’t seem to take normal wear and tear into account.

We’re in England, London to be exact.

My main question is: if we surrender possession of the property, can they legally make us continue paying rent for a period when the property is vacant because they’ve chosen to leave it empty before the next tenant moves in? Or should our liability end either when we surrender the tenancy or when a replacement tenant is available?