The Cumbrians made it three wins in a row under Kavanagh as they came from behind to beat struggling Notts County 2-1 at Brunton Park.
Callum McGregor fired the Magpies ahead in the first half only for Matty Robson to equalise with a clever free-kick.
David Amoo's sixth goal of the season settled it after the break leaving County still in search of their first away win
However, in the end the hosts had goalkeeper Mark Gillespie to thank for a series of magnificent saves to deny Chris Kiwomya's team who threw everything at the home defence in the last half an hour.
"I'm hoping they offer me the job," said Kavanagh. "If they want to be reminded of three wins from three I will shout it from the rooftops.
"County were different class and caused us problems but if you play with intelligence, honesty and hard work you get results. We weren't fluent but it was great play by Notts County.
"You earn your luck by hard work and three wins is a fabulous feeling. This is the day I feel most exhausted and I would not like to see the stats of the game.
"They were very good and I'm sure they will turn it round. I told the players they didn't deserve to win, but we will take it anyway."