To fuck. "A kick at the cat": a fuck. "Cat" from "pussy" and its furry nature, and "kick" probably because of the similarity of the thrusting action to violent kicks.
Dude, I'm so dyin' to kick the cat tonight!
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"Mard" is an old British circumlocation for "shit" that comes from the French "merde" = "shit". As used nowadays as a noun it means a sulky, petulant, ill-tempered attitude, or a person with that attitude.
In that use it's probably a contraction of "mardy bum" (= "shitty ass"), a synonym used in the Arctic Monkeys' song of the same name.
Well, now then, mardy bum
I've seen your frown and it's like looking down
The barrel of a gun.