This was a black and white "B" or "C" film of the 40s or 50s.
It was known for introducing a song that became a jazz standard.
A 25 to 30 year old man is singing a song to a 10 to 12 year old girl while riding in a horse carriage that he is holding the reins.
If the horse carriage was a car, he's on the passenger side and she is on the drivers side. I am pretty sure that the male actor and the little girl were not well-known in films.
The camera angle is facing them and no horses can be seen.
He is not dressed as a cowboy, he is dressed in a sports coat and tie of that era.
This is not a live shot, it's obvious that they are in a studio and the background was projected on a film screen.
Prior to that scene, one of those old-style captions pops up on the screen with the text... "Once upon a time, a long time ago in 1946..." or something similar to that.
The girl is initially sad but after the song she becomes happy.
I remember that the male actor didn't do a great job of singing that song.
The male actor's dialogue after the song goes something like this, "You're just jealous... ha ha, oh boy it must be tough to be a girl. How about I take you to [some event] to make you happy". Her response, "You're just saying that because you want to see [some girls name]". His response, "That's not true, why would I have spent that money buying you [something]".
I never saw the movie but I saw this scene on Youtube a while back and I am pulling my hair out trying to find it again.
As you can see, I can remember a lot about the scene but I can't for the life of me remember the song, the actor nor the movie.