The defender appeared to retaliate with a swinging elbow to a challenge by Adam Murray and County Durham referee Nigel Miller gave him a straight red.
It was one of the less controversial decisions by the man in black, who also booked three players from each side in sometimes puzzling circumstances.
In pouring rain, Carlisle fans were left whistling anxiously for the end after the Cumbrians had let Peterborough take the second half initiative.
Hawley squandered a golden chance in just the fourth minute - scooping a first time shot wide from Brendan McGill's pin-point cross.
And Kennedy blocked a fierce Peter Murphy drive on the six-yard line as Carlisle continued brightly.
They lost big front man Derek Holmes with a cut head in the eighth minute after an aerial clash with Sean St. Ledger, with Raphael Nade coming off the bench to replace him five minutes later.
Then a well worked set piece unlocked Peterborough three minutes later. Dean Holden brought down Hawley on the corner of the box and Adam Murray floated a precise free kick towards the far post, for Hawley to meet it and flick a back header beyond Mark Tyler's reach from eight yards.
Reduced to ten men, Posh brought on defender Chris Plummer for winger Ryan Semple and played with three at the back.
And they came close to leveling four minutes after the break when Mark Arber arrived late and unmarked to meet a far post Paul Carden corner but headed it into the ground and the ball bounced over.
Keeper Tyler scrambled a curling Murphy free kick past the post in the 53rd minute, before another Carden corner found Arber again but this time Williams saved his on target effort.