Puel has faced pressure in recent weeks and though it was eased by impressive festive wins against Chelsea and Manchester City, the 2-1 defeat to League Two minnows Newport will do him no favours.
The Frenchman was roundly jeered by Foxes supporters after the game as he passed them to conduct his post-match media duties.
Padraig Amond's 85th-minute penalty secured Newport's first FA Cup win over top-flight opponents since 1964 and started the inquest into Leicester's latest cup blow.
England internationals Jamie Vardy, Harry Maguire and Ben Chilwell were among those missing as Puel made seven changes from the New Year's Day win at Everton.
"I think we had enough quality on the pitch to qualify," said Puel, who was sharply criticised for naming an under-strength side for the Carabao Cup quarter-final defeat to makeshift Manchester City opponents last month.
"I don't want to look for excuses about this. I think we lost together, we are all together disappointed and frustrated after this game, because it was an objective for us. This competition was important."