"My feeling at the start of the season was it was going to be a hard graft," Moyes told BBC Radio.
"But ultimately today I feel for the supporters. They pay their hard-earned cash to come to the stadium and we have not given them enough this season," he said.
Moyes said he was not ready to discuss his future, saying two or three weeks of reflection were needed for the club before a plan was developed for a return to the Premier League.
The club's American owner Ellis Short issued a statement saying such a plan would be developed once the season was over and admitting the club had made mistakes.
"I acknowledge that during my ownership mistakes have been made, particularly in the area of player recruitment, and as a result we have found ourselves struggling to survive in recent seasons," he said.
"We had massive disruption during the summer transfer window and an unprecedented number of injuries throughout the season. These are difficulties which we have been unable to overcome and we are paying the price for that now."
At the Hawthorns Jamie Vardy struck the 43rd-minute winner for champions Leicester against West Brom in a 1-0 win.
Vardy finished with confidence after Shinji Okazaki latched on to a poor back pass from Salomon Rondon.
Goals from Ashley Barnes and substitute Andre Gray helped Burnley to get their first away win of the season with a 2-0 victory over Crystal Palace. The result moves Burnley to eight points above the relegation zone and should secure their top-flight status for another season.
Burnley move up to 14th place, above Palace and West Ham, who drew 0-0 at Stoke City.