LONDON (Reuters) - British police said on Friday there had been almost 6,200 hate crimes reported in the last month following the vote to leave the European Union in a referendum where immigration had been a key issue.

In the four weeks from June 16, police forces across the country said 6,193 offences had been reported, with the most common crimes being harassment, assault and other violence such as verbal abuse or spitting.

Britons voted on June 23 to exit the EU following bitter and deeply divisive campaigning in which the control of immigration was one of the main arguments of those who supported leaving the bloc. Since the result was declared, Muslims and Eastern Europeans say they have been particularly targeted.

The latest figures showed there were 3001 offences in the first two weeks of July, down 6 percent compared to the previous fortnight but still 20 percent higher than the same period last year.

"Following increases in hate crime seen after the EU referendum, police forces have been taking a robust approach to these crimes and we are pleased to see the numbers of incidents have begun to fall," Mark Hamilton, the National Police Chiefs' Council spokesman on hate crimes.

"Clearly any hate crime is unacceptable and these numbers are still far too high."

Critics of the "Leave" campaign say its focus on immigration helped stoke xenophobia and racism, an accusation its leaders reject. A week before the vote, opposition Labour lawmaker Jo Cox, a strong supporter of remaining in the EU, was shot and stabbed to death in her constituency in northern England.

(Reporting by Michael Holden; editing by Giles Elgood)