Martina was brought to Everton by Allardyce's predecessor Ronald Koeman in July.
Everton fans may have witnessed their side's biggest win of the campaign in any competition - against a West Ham United one managed by their old boss David Moyes, no less - but that hasn't spared one Toffees player from criticism on social media.
Wayne Rooney stole the show at for Everton, scoring a hat-trick of goals including one from inside his own half, before Ashley Williams' header finished things off in David Unsworth's final game in temporary charge before handing over the reins to Sam Allardyce.
Everton were not always comfortable, mind, and found themselves up against it early in the second half, at which point West Ham's Manuel Lanzini saw his penalty saved by Jordan Pickford.
And one Everton defender in particular would be best advised not to log into Twitter this morning...
Please let tonight's great result not make everyone forget that Cuco Martina needs reporting to the UN for crimes against football. One of the worst players I've ever seen in an Everton shirt, a better team would have had a field day against us him tonight. Get in the bin. #EFC— Paul McAllister (@p_mcallister123) 29 November 2017
Leighton Baines might only be half the player he used to be but he is still 10x the player Martina is! Hurry back Bainesy! #EFC— Liam (@James47Liam) 29 November 2017
Poor start to 2nd half Everton. Martina nearly cost us that goal! Doubt he will ever play again under Big Sam. He’s garbage. #EFC— Morgan (@MorganEFC) 29 November 2017
Poo West Ham, where that bad even Cuco Martina had a bang average game— Mark Corbett (@MarkCorbett87) 29 November 2017
On a serious note though, Cuco Martina needs to get in the bin.— Joey (@JoeShaw254) 29 November 2017
How can a professional footballer not control the ball with his left foot. Cuco Martina still sat in left back though eh— Michael Thomson (@mthomson25) 29 November 2017
Cuco Martina at right back is bad enough. Left back is like he’s playing a different sport.— Smithy (@ijjysmith) 29 November 2017
Cuco Martina playing left back is making me unwell— Saul Anderson (@SaulAnderson1) 29 November 2017
Martina, who was brought to Everton by Allardyce's predecessor Ronald Koeman in July, was making his first league start since 1 October.
And in his defence, he was having to cover for Leighton Baines on his unfavoured left side.