Mick Creedon, Derbyshire's police chief constable, said the law, which comes into effect today, will be given a "very low" priority, blaming cuts to force figures and under-resourced traffic units.
The crackdown is designed to protect children from the effects of tobacco smoke, seeing drivers and any smokers fined £50 if they have someone under 18 in the car too.
But speaking on Thursday, Creedon said police would be forced to tackle other, more dangerous crimes being committed on the roads, including drink driving.
"As we're losing resources hand over fist, the few traffic police I have will be concentrating on the issues which are to do with death and injury on the roads," he told BBC Radio 5 Live.
"My other staff will be looking at the risks about child protection, organised crime, terrorism and the other calls we get day in day out. I am not remotely against the law but as a policing priority, it'll be very low.
He said staff were not tasked with driving around looking for issues "like this".
"Mobiles, drink driving, speeding, no seatbelts - we know these contribute to fatalities on the road. Smoking in a car doesn't bring the same danger on the roads," Creedon added. "It's a public health issue."
His remarks come as new research shows the harm secondhand smoke can pose to youngsters - even if car windows are wound down.
Anil Namdeo, of Newcastle University's Transport Operations Research Group, carried out experiments to test levels of dangerous chemicals which children sitting in the back of a car would breathe in.
His team tested having the windows open or closed, fans on or off and with the air recirculating or not.
Drivers replicated a school run in Newscastle, using volunteer smokers. Dummies were used in the back as no children were involved.
There evidence proves that despite common misconception, with the window open, toxic chemical levels were more than 100 times higher than recommended safety guidelines.
And with windows closed and the fans on, levels were more than 200 times the safe limits.
Levels of poisonous carbon monoxide were two to three times worse than on a busy road at rush hour.