British police on Friday said they recorded a 42-per-cent jump in hate crimes over a two-week period in June, the month in which a migration-focused campaign on EU membership culminated in the country voting to leave the bloc.

Police forces nationwide recorded 3,076 "hate crimes and incidents" from June 16 to June 30, up 42 per cent on figures for the same period of 2015, the National Police Chiefs' Council said.

"We now have a clear indication of the increases in the reporting of hate crime nationally and can see that there has been a sharp rise in recent weeks," said Assistant Chief Constable Mark Hamilton, the council's lead officer for hate crime.

The reported incidents reached a daily peak of 289 on June 25, the day after the result of the Brexit, or British exit, vote was announced. The council said there had been "a marked decrease" in reports since then.

But police and local media have reported more apparently Brexit-inspired incidents of alleged hate crime this week, including threats against at least two black politicians.

In one of the most serious incidents, police said they were investigating an arson attack targeting a Polish family in the south-western city of Plymouth late Wednesday.

"Soon after the fire was started, the occupants of the property who originate from Poland, discovered that a hate-filled letter had been sent to them, containing further threats," Devon and Cornwall Police said.

The letter told the family to "go back to your country," the local Plymouth Herald reported.

"We believe this fire was started deliberately, was racially motivated and could have endangered the lives of the occupants," police inspector Darren Green said in a statement.

Another arson attack was reported at a halal butcher's shop in Walsall in the West Midlands last week.

Nazir Ahmed, a Muslim member of the House of Lords, the upper house of parliament, received an envelope containing a racist note and non-toxic white powder on Thursday. The note included the words "Paki filth," Ahmed told the BBC.

At least three mosques in London received similar messages in envelopes containing white powder, local media and monitoring groups said.

"Sadly, we can confirm that some mosques and Islamic centres have received racist and anti-Muslim comments with white powder attached," the group Measuring Anti-Muslim Attacks, also known as Tell MAMA UK, said on Friday.

Pro-EU politician David Lammy said he contacted police earlier this week over racist abuse and threats following his calls for parliament to vote on Brexit.

Lammy, a black member of the Labour opposition in parliament, said he had "received a barrage of abusive, racist and threatening messages" since the referendum.

Tell MAMA said the recent incidents painted a "troubling picture of what is taking place in our country."

"Racial, cultural and faith-based hate have been triggered, and politicians involved in Brexit have left ... the police and communities to pick up the tab," the group wrote on Twitter.

Shuja Shafi, head of the Muslim Council of Britain, which represents some 2.7 million British Muslims, also warned that the recent rise in reported hate crimes suggested that "our country is experiencing a political crisis which, I fear, threatens the social peace."

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