The boy, aged five, was reportedly hit by an e-scooter ridden by a man in a pedestrian area and needed hospital treatment to his face and legs.
While people can legally buy an e-scooter, it is against the law to ride their own scooter on the road, cycle lane or pavement.
If a person is stopped by the police, officers could seize the e-scooter. They could also get a fine and points on their driving licence.
A police spokesman said: “Our colleagues at Safer Roads Humber are continuing to remind people of the law surrounding the use of e-scooters and we are reminding people of their statement. Whilst you can legally buy an e-scooter, it is against the law to ride your own scooter on the road, cycle lane or pavement.
“If stopped by the police in these areas, they could seize the e-scooter and you could get a fine and points on your driving licence.
“Parents – if you have bought an e-scooter for your child at Christmas please make sure they remain within the law.
“Be aware that privately owned scooters are restricted for use solely on private land with the permission of the land owner.
“It is illegal to use a privately owned e-scooter on pavements, cycle paths or roads.”