Brenford forward Marcello Trotta has been put forward for the League One Player of the Month for January.
Posh striker Ajose spent the first half of the season out on loan with Swindon Town but has returned with a bang, scoring five goals in as many games that included a hat-trick in the 4-3 victory over Notts County.
Orient won all four games in the opening month of 2014, letting in just one goal along the way, with Cuthbert also scoring two goals from defence.
Another loanee up for the prize is Newcastle United's Tavernier, who had to play at both right and left-back for playoff-chasing Rotherham this month.
He also played a part in all three goals in the victory at Shrewsbury Town, setting up two and scoring the other himself.
The winner will be announced on Friday.