Former Republic of Ireland midfielder Kavanagh, on the coaching staff at Brunton Park since 2008, was placed in temporary charge when Abbott was sacked last week and made his intentions clear after the Cumbrians' 1-0 home win over Sheffield United.
"I do want to say a huge thank you to Greg," Kavanagh said. "I've spoken to him for advice a couple of times this week and that's the type of man he is.
"He's at the end of the phone when he could quite easily blank me. He's not that type of character and all he ever wanted was to do his best in very difficult circumstances when he was here.
"I think we now need to get away from the negativity and the difficult circumstances and make the place as positive as it can be.
"Let's try and get this juggernaut rolling again in the right direction.
"I do want to take it forward from here and I would love to be given the opportunity to do that."
The club's board confirmed last week that they wanted a new manager in place in time for Saturday's league game at Stevenage.
Danny Wilson, former Carlisle midfielder and manager Paul Simpson have emerged as favourites for the job, while Peter Schmeichel has also been linked with the vacancy.