A brace for Sean Brain gave City a first win at Claines Lane in the Preliminary Round of the FA Cup.
His first came in the 17th minute, firing low past Wren after a scramble from Jamie Smith’s cross into the middle.
The lead was doubled soon after when Brain got in behind the defence and slotted past keeper James Wren
Wood responded by making two substitutions which steadied them down and they responded early in the second half when Michael Nelson crossed to the back post for Kyle Rowley to power a header into the roof of the net.
Reynolds came close for City seconds after the restart but his shot was blocked from close range. Lewis Hayden then came close for Wood but his volley just cleared the crossbar. Goalkeeper Wren nearly snatched the equaliser as he went forward in added time but his header went inches wide.
The win sees City go into the draw for the 1st Qualifying Road.