They host second-placed Ealing, who still hope to chase down Championship leaders Bristol, at Emerald Headingley tomorrow.
New Zealander Mayhew knows Carnegie fell 44-19 in the reverse fixture earlier this season but believes his side have improved since.
Furthermore, bottom-placed Rotherham Titans almost stunned Ealing last week before eventually agonisingly losing out 37-35.
“We have shown that when we get our first-choice team out there, we can match anyone,” said flanker Mayhew, who this week became the first Carnegie player to re-sign for 2018-19 on a one-year deal.
“Hopefully we can show that this Sunday against Ealing, as they are right up there.
“I enjoyed playing them last year and they have kicked on from that.
“They have made a lot of signings for next season and shown their intent to improve again.
“We back ourselves and we need to repeat what we did against them last year (in the Championship semi-finals)
“We will need to be physical and be accurate at our set-piece; that will give us an opportunity to beat them.
“We know it will be tough but we want to keep our good form at home up for the remainder of the season.”
On his new deal, the ex-Newcastle Falcons star admitted: “It is really pleasing to have it all sorted.
“I didn’t want to leave the club and it is great to get it done so I can look forward to finishing off this season well and building for next year. This year we have not performed at times as well as we would have liked to.
“We are still sitting in fourth but I don’t think we have played to our potential yet and we are due a big game.
“This is a great opportunity in the next two games – Bristol away next Friday – to do that against the two strongest teams in the Greene King IPA Championship so far this season.
“We are looking forward to it and are pretty pumped for a big game on Sunday,” added Mayhew.
Carnegie have made five changes to the side that lost to Bedford Blues including the return of full-back Chris Elder, lock Ben West and flanker Josh Bainbridge but Mike Myerscough and Trystan Roberts drop out with injuries.