Yes, but the need for a present perfect would rarely be justified by context. The 'tyre' is sometimes omitted: 'I had a flat on the way here' is possible. You can also say 'blow-out' (if it's a dramatic and sudden deflation). 'Blow-out' can also be used, though there's another meaning for 'blow-out' - to do with eating a lot*. But if you turn up late at a party and say 'Sorry I'm late, I had a blow-out on the way' there's little risk of confusion. ;-)
b
PS * The film La Grande Bouffe was translated as The Blow-Out.