The fly-half endured a nightmare with the boot at Twickenham, missing three penalties and a drop goal to let France off the hook after England had led
17-6 at half-time.
"I'm very disappointed with the result and with myself," Hodgson told BBC 1.
"It is very hard to take but it's something I will have to get through and come back stronger.
"All week training and practice has been good but, when it came to the game, it just didn't happen for me."
Hodgson, who revealed that centre Olly Barkley had taken three penalties because they were "out of my range" was dismayed at missing the drop goal that would have got England out of jail towards the end of the match.
"It wasn't a good strike," he groaned.
"I felt as soon as it hit my boot that it had missed.