Cole made a bright initial impression against Rabotnicki in a Europa League qualifier but that was quickly forgotten as he was sent off on his first Barclays Premier League appearance for the Reds against Arsenal.
That incurred a three-match domestic ban but, free to play on in Europe, he compounded his misery by missing a penalty against Trabzonspor on Thursday.
But Hodgson said: "The biggest blow is for him. He was so desperate to start his career here well and he hasn't done so.
"In his first game he managed to get himself sent off - albeit somewhat harshly - and in his second he missed a penalty.
"It has been a bad week for him and now it is going to be even worse because after the game in Turkey he will have to sit on his backside for two or three weeks because of the international break.