We went last year, once in mid April, than again around the end of May. We did see a few puffins in April but more in May, I'd also say have a walk from North Landing to Flamborough Head, there's quite a few outcrops to walk out on,check out Google earth, looking back and down we saw more puffins there.
I came away with quite a few photos of Gannets, Guillemots and Razor Bills I was pleased with, but to be honest, found 300mm (thats the longest focal length I have) a bit short for the puffins.