Can a tenant avoid small claims court by part payment

Tenancy Types and Management

I have asked my letting agent to recover service charges owed by my tenant. The tenant owes £6000 so the agent has lodged a claim online. As the claim is above £5000, this should be outside of the small claims court procedure and I can ask the court to order that the tenant pays my legal costs. What happens if the tenant is served with notice of proceedings and pays off say £2000 of the claim after service of my claim for £6000 as an undisputed amount? Will the claim then become a small claim and prevent me from having my legal fees paid by the tenant??