A section of the Carlisle crowd called for Abbott's head at the end of a game, which could easily have ended with greater embarrassment for the Cumbrians.
Carlisle have leaked 13 goals in their first three Sky Bet League One games and it was clear that the patience of their supporters was wearing thin as the home defence was cut apart by a rampant Sky Blues' attack.
But Abbott pledged not to give in and said: "It's one of the toughest days I've had in football.
"It was embarrassingin the end and the dressing room is so fragile. Every little thing affects the players and I feel as a group we have let people down.
"The game was lost within five minutes, but we are better than that.
"The early goals killed our confidence. I'm not going to give in. I will take the bullets and fire some back when we deserve it.
"I'm not sure we have been as desperate as this in my time here.
"It has hurt like mad and if we don't put it right we will be relegated."
Leon Clarke's 20-yarder and a solo effort by Franck Moussa set Coventry on their way inside a torrid opening six minutes for the home side and Billy Daniels picked his spot to make it 3-0 before half-time.
James Berrett missed a penalty for Carlisle and Clarke profited from Kevin Feely's mistake to add number four in the second period.
City should have had more but for a series of fine saves by Carlisle goalkeeper Mark Gillespie, while the woodwork robbed ex-Carlisle striker Mathieu Manset of a goal on his return.