The incident happened at around 9.55am on Wednesday (August 27) at the pelican crossing on the junction of Lendal Bridge and Station Avenue.
The little girl was walking with her childminder and an assistant when she began to cry.
A man standing in front of them turned around, walked a couple of paces and kicked the toddler in her stomach. She suffered a minor external injury as a result of the incident.
A woman who was with the man immediately apologised for the incident explaining that the man had a behavioural condition.
Police are appealing for the man and woman to come forward and to anyone who witnessed the incident to contact them.
The man is described as aged between 20 and 35, of heavy build, approximately 5ft 6in tall with thin hair.
He was wearing dark coloured trousers and an older style jumper with stripes. His shoes are described as big and he appeared to be dressed in clothing which an older person would wear.
The woman who accompanied him is described as around 5ft 6in tall, aged in her early twenties, of slim build with dark hair.
Police are asking anyone who saw the incident to contact them. They are also appealing for the woman who was with the man, or anyone who can help identify the pair, get in touch as soon as possible.
Please contact North Yorkshire Police on 101, select option 2 and ask for PC Craig Appleby. Or email [email protected].
If you prefer to remain anonymous, you can pass information to Crimestoppers on 0800 555 111.
Please quote reference 12140146357 when passing on information about this incident.