Kavanagh, 35, has made 13 appearances for the Cumbrians since arriving in October, but is due to return to parent club Sunderland following Saturday's League One meeting with Leicester City.
Abbot does not believe Roy Keane's decision to step down as manager of the north-east side last week will affect his side's hopes.
"I have had talks with their caretaker manager Ricky Sbragia and it's looking good," he told the News and Star.
The former Republic of Ireland international has not made a first-team appearance for the Black Cats since they won promotion to the Premier League two years ago.