Quality opportunities were at a premium before the break as neither side managed to make the most of a string of half chances.
The Robins took control after the break as they created a string of chances with Paul Hartley volleying wide before Marvin Elliott’s header was saved by Arran Lee-Barrett.
Haynes came agonisingly close to scoring in the 53rd minute as his effort from the edge of the area came back off the bar.
The hosts continued to search for the opener with Elliott denied by Damien Delaney’s block before Maynard failed to make the most of a close-range header.
The point sees Ipswich remain 22nd despite extending their unbeaten run to eight league games, while City climb to 11th.