Sam Allardyce’s side have slipped down the Premier League table to 10th place after a run of just one win in 11 games.
Their latest disappointing result came at Manchester City on Sunday as they lost 2-0 and they are now completely out of contention for a European berth, while relegation is not an issue.
Despite the Etihad reverse, Jenkinson said: “People in our dressing room are fired up. If you came in the changing room after the game you’d see that.
“People aren’t resting on their laurels, people aren’t happy finishing mid-table. We want to kick on. We have got something to prove now because we have been on a bad run of games and we want to put that right. We owe that to ourselves and the fans.”
Jenkinson felt West Ham paid the price for their mistakes against City and does not believe the team are much doing much differently from the early part of the season.
He said: “I wouldn’t want to blame it on luck but a few things haven’t gone our way lately and we have conceded some late goals.
“I think this was a different story – we made a couple of mistakes and got punished.
“Of course we have got to work on our performances but if you look at the last 10 games, our performances haven’t been bad at all.
“We have had some really good performances but teams have scored late goals against us and we have ended up getting a point instead of three. It has been a bit frustrating.
“I just think early on in the season we were finishing everything. Everything we got was going in the back of net and at important times of the game.
“We have gone ahead in recent games but never killed the games off and have ended up conceding cheaply towards the end of the game. Of course that is a real kick in the teeth.
“It is not nice blaming things on luck – you make your own luck – but that has been the story of our last couple of months.”