The full-back has enjoyed an impressive campaign on-loan at Upton Park from Arsenal, but the Hammers' promising form has tailed off in recent months as one win in 11 league matches has seen them slip from the European places into mid-table.
Their latest setback came on Sunday with a 2-0 defeat away to Manchester City, but with five games remaining, England U21s defender Jenkinson insists he and his team-mates have something to prove.
'People ain't happy finishing mid-table - we want to kick on'
"People in our dressing room are fired up. If you came in the changing room after the game you'd see that," said the 23-year-old.
"People aren't resting on their laurels, people ain'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."
West Ham were in the lofty position of fourth earlier in the season, but a combination of bad luck and a lack of ruthlessness has behind their drop in form according to the former Charlton Athletic man.
"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," he added.
"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," he added.
See Carl Jenkinson's goal for England U21s against Portugal earlier this season
"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."
Jenkinson will be hoping that his performances between now and the end of the season will earn him a spot in Southgate’s squad for this summer’s showpiece in Czech Republic.
The U21s boss will name his 23-man group for the tournament next month.
