Nine police officers were injured during an anti-lockdown protest in Bradford, police said.
Up to 400 demonstrators marched through the city centre.
West Yorkshire Police said the "majority of protesters were peaceful", but added a group had to be stopped after they tried to make their way to a vaccination centre.
Police arrested 13 people for public order offences and issued 19 fines for breaching Covid regulations.
Temporary Assistant Chief Constable Oz Khan, of West Yorkshire Police, said policing the demonstration was "challenging and complex".
"We are grateful to the communities of Bradford, who have attended the protest in a peaceful way," he said.
"However, there was a group of people who had travelled from outside the Bradford District to cause trouble.
"There was a small group of protesters who tried to break away from the protest and make their way to a vaccination clinic. Officers intervened to stop this from happening.
"The majority of people were peaceful, and I am pleased to say there was no significant disorder."
In Sheffield, police said about 1,000 people attended a protest in the city centre against the government's controversial new police and crime bill, which has seen rallies held across England.
Ch Insp Paul Ferguson said: "No arrests were made and those who attended did so peacefully."