A 32nd-minute strike by Andrew Crofts sent the Royals diving to their fourth defeat in the last five matches as Brighton clinched a 1-0 win at the Amex Stadium.
former Saints boss Adkins felt speculation linking him with the West Brom job had simply come from bookmakers shortening the odds.
He said: "The bookmakers are covering themselves and this has come out of the blue. West Brom don't have a manager but all my focus is to try and improve Reading. All the energy and focus is to us.
"I wanted a good cup run but unfortunately a deflected goal has done us today."
If Adkins was appointed this week, his first game would be against Saints at The Hawthorns next Saturday.
Reading have only scored five goals in their last eight games and Adkins believes the club have no option but to be active in this month's transfer window if they want to have a successful second half of the season.
He added: "We've got a fantastic board and the chairman will do everything that is right for the club. We've got to let the board get on with it.
"We've got an opportunity to regroup and maybe get one or two in because our focus is to get promoted."
Adkins had taken his former Saints striker Billy Sharp on loan for the first half of the season, but the player has now returned to St Mary's where he is obviously out of favour.