Toilet difficulty?

Hi all,
I have an odd problem with our toilet. Lately, I've been needing to repeatedly pump the lever to get it to flush properly. A normal push of the lever would produce just a small leak instead of the regular 'full' flush. Any thoughts at all on what that might be?
I've taken the lid off the tank and inside is a plastic contraption that has a flap inside of it that seems to operate the flush, however the flap moves correctly when the lever is pushed.

You've described the standard symptoms of a torn or perforated diaphragm.   

The part you can see lifting inside the siphon assembly isn't a solid disk but basically just a supporting grid on top of which a thin plastic diaphragm sits and is anchored at its center.  When you flush, the diaphragm lifts the water above it, and as the water is forced into the narrower section and over the top of the assembly into the flush pipe it starts a siphoning action, the self-sustaining flow of water then causing the diaphragm to lift upwards away from the grid.  When the cistern is almost empty, air entering the bottom of the assembly breaks the siphon action, and the diaphragm drops back down ready for the next operation. 

As the diaphragm gets larger tears or holes in it, it takes more and more effort to start the siphoning action, eventually reaching the point where you have to pump it up and down quite rapidly followed by complete failure.

Time to fit a new siphon assembly:
http://www.screwfix.com/prods/44047/Plumbing/Toilet-Fittings/Toilet-Siphon-203mm-8

I'll put that one in one of the few remaining cells of the ol' grey matter

Hell of a lot cheaper than a plumber!