Jones appeared to have bundled the ball over the line - and the linesman thought so, too - but referee Brett Huxtable did not signal a goal until Wyke smashed it home after goalkeeper Luke McGee had clawed it back into his path.
However, the Devon official emerged to confirm that he had awarded the strike to Jones, and it was enough to give City victory after three successive draws and keep them firmly on course for the play-offs.
Bradford suffered a pre-match blow when midfield driving forced Nicky Law was ruled out after taking a knock to his calf late on in the midweek draw against MK Dons.
Better news was that Mark Marshall had fully recovered from hamstring trouble, enabling Stuart McCall to name two wingers with Newcastle loanee Alex Gilliead on the other flank.
The other change saw January signing Jones, who has been dogged by hip problems, replacing Billy Clarke to play alongside deadline-day buy from Carlisle Wyke.
Posh, who had won on their last two visits to Valley Parade, went 4-4-2 and had Craig Mackail-Smith and Tom Nichols up front.
They made the better start attacking the Kop through the direct running of winger Marcus Maddison but it was City’s Marshall who forced the first save with a stinging effort from the corner of the area.
Rory McArdle also headed wide of the far post from Tony McMahon’s corner kick before Paul Taylor hooked over from a quickly-taken corner for the visitors.
City went ahead in the 24th minute when Josh Cullen teed up Marshall to hit a low drive which McGee failed to hold. In a mad scramble, Jones appeared to bundle the rebound over the line but McGee pulled the ball back and Michael Bostwick knocked it into the path of Wyke, who made sure by smashing it into the net.
Maddison bent a free-kick into the side-netting after Nat Knight-Percival went into the book for bringing him down and Nichols also went in the book for handling before the break.
Posh should have been level in the 48th minute when Mackail-Smith cut in from the right and squared for Nichols, who saw his low drive blocked by the sentry that was McArdle eight yards out.
A mix-up between McGee and Anthony Grant also resulted in an own goal as City hit back.
Goalkeeper Colin Doyle was relieved to see a shot on the turn from Nichols curl narrowly wide before Jordy Hiwula replaced Jones in the 63rd minute for City.
Michael Bostwick went in the book for bringing down Hiwula on the left corner of the area but City could not capitalise from McMahon’s free-kick which forced a corner off the three-man wall.
Marshall fired over after the Kop bayed for a penalty for hands by Ryan Tafazolli.
Posh had used all three substitutes by the time Bostwick headed over a 76th-minute corner from Maddison.
Marshall gave way to Timothee Dieng for the closing stages and the substitute should have sealed it when sent down the left by Cullen only to curl his shot over the bar.
McGee raced up in vain but City repelled a stoppage-time corner and claimed the points.