The 54-year-old broadcaster was on air on the station on Wednesday following reports that he had been axed.
But returning to the airwaves, Wright admitted his future was uncertain.
He told fans: "Make the most of it, folks. We could be dragged out of here at a moment's notice!"
Wright admitted that he did not know what "is going on, truth be told, whether it be the General Election or Brexit or my own future".
He added: "It was all a bit chaotic last night ... I turn up for work today as I was expected to do.
"Guess what happened ... We were told we were going to be signing new contracts.
"So we've not been fired with immediate effect ... We're not going our separate ways as I thought might happen at Christmas," he said of co-host Kevin O'Sullivan.
"We're here into the new year and maybe even beyond.
"There's never a dull moment on TalkRadio."
He thanked his fans for their messages of support, saying: "Really, thank you a lot."
And he tweeted: "Thanks again for all the kind words - I've had a tough time with my mental health these past 18 months.
"I was expecting to leave TalkRadio at (the) end of Xmas but they'd like me and Kevin O'Sullivan to stay on longer now - thanks again."
Wright previously revealed that the stress of presenting his long-running Channel 5 show had left him with symptoms of post-traumatic stress disorder (PTSD).
He later told listeners: "I must admit when I got up this morning I thought I might be spending the rest of my life on holiday - low on sunshine, high on relaxation, low on money!"
And when a producer spoke to him off-air, Wright quipped: "When I get a voice in my ear I think 'This is it, they're coming to get me, we'll have to down tools'."
After fitting his own dishwasher, he said: "I was thinking of new career paths last night ... I could have a second career as a dishwasher fitter!"