Club captain Chris Read hit 88 as Notts reached a second innings total of 279 on day three at Trent Bridge.
Play closed with Kent 129-4, needing 115 runs on the final day to claim victory.
Read’s knock, which included a partnership of 67 with Mark Ealham, provided stability after the fall of Voges, Patel and Wood, all caught behind by Geraint Jones off Ryan McLaren, the pick of the Kent bowlers with 6-75.
Stuart Broad contributed 20 but Ryan Sidebottom was seen off having scored a solitary single before being caught by Martin van Jaarsveld at slip after eight balls.
Broad enjoyed a fine spell with the ball, taking 3-27, his victims including Kent openers Joe Denly and James Tredwell.
“Stuart would have been a bit disappointed yesterday but he was excellent today,” said Notts Director of Cricket Mick Newell.
“We lost too many wickets too quickly in the middle order and there’s no excuse for that because the ball didn’t do a heck of a lot.
“Chris’s knock has given us a chance and but for that it could have been all over but we’ll come at them hard in the morning and see where we get to.”