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The result means Worcester sit 19th in the Southern League Premier Central table, three points from safety in the relegation zone.
Banbury created the game’s first chance at Gustavo Sanches fired narrowly wide of the post on the turn.
The visitors felt they should have had a penalty just past the quarter-hour mark when Liam Lockett went down under a challenge in the box but the referee waved away City’s appeals.
However, City broke the deadlock on 25 minutes when Wise headed in at a corner to put the visitors ahead.
A Worcester player then fluffed his lines just past the half-hour mark with only Banbury goalkeeper Jack Harding to beat.
Banbury responded with a couple of chances of their own as the visitors put their bodies on the line before Cristiano Dzialuk dealt well with a corner.
City took a one-goal lead into the break and doubled their advantage just four minutes after the interval as Bishop reacted quickest to tap home a loose ball in the box and mark his debut with a goal.
Worcester saw a second penalty appeal turned down by the referee past the hour mark after Idehan felt he was pulled down at the back post at a corner.
However, Idehan sealed the points for Worcester on 69 minutes when he headed in from point blank range at the back post after City flicked on a corner from the left.
City almost made it four soon after as Banbury blocked Josh Taroni’s effort to safety before Levi Steele headed over from close range for the home side late on.
Worcester are next in action against league leaders Real Bedford on Saturday, October 31 with kick-off at 3pm at Sixways Stadium.
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