The tread in the photo is tapered towards the left hand side, caused by hard cornering on track.
You don't necessarily want a tyre with a harder compound (you will see less grip with a harder tyre), you want a tyre that can hold its shape better under excessive force. Mainly, what you will need to look for is a tyre with a bigger contact surface like R888's, so stay away from tyres with an anti aquaplane V type tread pattern as these wont last long on track.
What you need to have checked is wheel alignment, paying extra attention to front toe and camber (caster doesn't really affect tyre wear too much), I would advise you only to have the alignment adjusted after new tyres have been fitted, old worn tyres will hide the true readings. Also you want to check bottom ball joints and track rod ball joints for horizontal and vertical play, If there is excessive play then this can put the alignment out under cornering and if left untouched can result in the ball joint separating, which wont be a very nice outcome. TBH tho, if the ball joints were that bad, you'd at least here a knocking or clunking noise when driving the car.