It would be a public service contract, not a public supply contract. Supply relates to the purchase (or hire etc) of items. If this is for award of a new contract (different rules apply to extending an existing contract) follow the paper trail from the model standing orders (Section 18) to the model financial regulations (Section 11) to the Public Contracts Regulations 2015 (Chapter 8).
Formal tendering will be required in compliance with your own standing orders/financial regulations and the opportunity must be advertised on the Contracts Finder website. If you do not have the necessary experience to do this, your local Association should be able to assist.
Visit the Contracts Finder website at https://www.contractsfinder.service.gov.uk/Search
and enter "grounds" in the search box. You'll see examples of similar contracts currently being tendered, including access to their tendering documentation, which will give you an indication of what is required.