* PM assumes responsibility for political turmoil

* Labour Party damaged in European election

* Far-right British National Party wins two EU seats (Adds quotes from Byers, Darling, Independent poll)

By Adrian Croft and Frank Prenesti

LONDON, June 8 (Reuters) - British Prime Minister Gordon Brown beat off a challenge to his authority on Monday, winning over Labour members of parliament after admitting mistakes and taking responsibility for a week of political turmoil.

Addressing Labour's 350 MPs a day after his party suffered a resounding defeat in European elections, Brown expressed contrition but said he was determined to fight on as leader after one of the most difficult weeks of his premiership.

"I know I need to improve," Brown told the MPs, according to a spokesman. "I have my strengths and I have my weaknesses. There are some things I can do well, some not so well.

"You solve the problem not by walking away but by facing it and doing something about it," he said, earning cheers and applause from the majority in the room, according to witnesses.

Several senior Labour members, including Charles Clarke, a former interior minister, called for Brown to step down. However, most threw their weight behind him ahead of a general election due within a year which the opposition centre-right Conservatives are tipped to win.