How can I make my petrol mower run better?

My Hayter Hawk 375 mower with Briggs and Stratton 4-stroke engine will start OK after priming, but only runs for a few seconds. If I re-prime it, the same thing happens again. This is the first trouble I have had of this kind in 10 years of ownership, and I was given it when it was already well used... How does the fuel delivery system work, and is this likely to be the source of my trouble?