Both West Yorkshire Police and South Yorkshire Police reported extra demand being placed on resources over the weekend, in particular on Saturday evening.
Officers warned people to not put those who are vulnerable at risk and to adhere to the rules with lockdown still in place.
Operations Superintendent for South Yorkshire Police Lynne Knox posted on Twitter: "Teams are being pulled in all directions with reported Covid breaches impacting our ability to respond as quickly as we can to help others.
"Please be responsible and help us protect those who need us most."
Steve Kent, chair of the South Yorkshire Police Federation, said: "This is putting an enormous strain on our resources.
"There are still Covid regulations in place and if people are breaking the rules it's down to our officers to respond and it puts an enormous strain on them because we are effectively dealing with two strands of incidents - those that are Covid related and other crimes such as domestic abuse.
"Covid breaches will never take the priority of emergencies such as domestic abuse, but it does put an extra demand n officers. Really, they should be of dealing with the serious crimes and making sure people are safe.
"We are still seeing a lot of the public acting as though we are nearing the end of restrictions but we are still under tight rules and the public have to abide by them otherwise the police will reluctantly have to intervene and enforce it."
The West Yorkshire Police Contact Management Centre also reported a busy weekend.
The contact centre's Twitter account stated on Saturday evening: "Lots and lots of Covid breaches tonight.
"Everyone knows the rule by now. Please stay home and stay safe, normality is close now."
Earlier this month, North Yorkshire Police Superintendent Mike Walker emphasised the importance of continuing to stick to the rules, following a significant upturn in the number of fixed penalty notice's handed out by officers.
He said: "We are not at the point where we can abandon social distancing and all the other precautions we’ve been taking over the past year and just throw caution to the wind.
"We will keep on policing the regulations and taking enforcement action against breaches, we will keep sending out patrols and we will keep engaging with the public , to protect all of our chances of moving out of lockdown as quickly and safely as possible."