He can promise £1,000 of his own money! No, individual councillors have no power to do anything. Council decisions can only be made in a properly constituted council meeting, where an agenda has been published three clear days ahead of time, and the items to be considered have been clearly stated.
The only exceptions to this are that council officers can make decisions subject to their delegated authority set by the council, and a council can delegate its functions to another authority (although this is rare).
Of course, a councillor could advise someone that in their opinion the parish council might look favourably on an application for a grant for a particular project. But that's plainly not a promise.