Clarke, 21, came off the bench at half-time in the 3-1 defeat to FC United on Tuesday night to make his Yellows debut.
He was handed his professional debut by now-Warrington manager Paul Carden at Tranmere Rovers back in 2016, and was released by Fleetwood Town in the summer, having spent spells on loan at Macclesfield Town and Stockport County.
Edwards, 34, has played more than 450 games in the Football League, notably for Accrington Stanley, Rochdale and Burton Albion.
The former Wigan junior was an unused substitute at Broadhurst Park in what was the last Yellows game before the enforced break due to lockdown.
The club have also confirmed the departures of Tom Warren and Chris Holroyd.
Carden said: “Sometimes there’s a fit and sometime’s there isn’t between a player and a club, and nobody signs with any intention other than to do well. I’ve known Chris a long time – since he was 19 or 20 – he’s a good lad but he needs to get a club that’s crying out for a starting striker. I just don’t think he was the right fit for us as it turned out, but those things happen.
“(Tom’s) on about going to America for a scholarship. He came back for pre-season and it just wasn’t quite happening for him. I expected a lot from him but again, these things happen.”