The letting agreement on my property end at the end of October and it is managed by a letting agent. After that I would like manage the property myself with the same tenants. The letting agent says I have to pay 1 months rent plus VAT. Is this corrent if the letting agreement ends anyway in October
This depends on the written agreement you have with the letting agent. If there is no written agreement then you may be able to dispte it.
If you have a written agreement but it does not say anything about having to pay any money when the agreement ends then the agent will find it hard legally to demand any money at the end of the agent letting agreement
will work out cheaper to self manage especially if you are a British landlord association member.
In Estate Agents term of business they do usually say that if tenancy is renewed with same tenants they are entitled to some fee. However, but some of the estate agents do not bother. However, in your case they are no longer going to manage the property so loosing their business so may demand it but it is usally £100 plus VAT or £250 fees not as much as 1 month. So please check your copy of term of business you have signed.