Great example of “fall behind.” I have one tip for you. There’s a big difference in meaning between these two things:

1. “…hard to see that my child is falling behind in school.” This means, it’s difficult for you to know that this is happening.
2. “…hard to see my child fall behind in school.” This means, as a parent, it causes difficult emotions when you see your child fall behind.

I think you were going for meaning 2, so I’d say this: “As a mother, it’s difficult to see my child fall behind in school. I’ve learned not to worry too much about this, as I know that almost all children catch up later on in school or even in their professional lives.”

Great example! Are you watching the first matches of the year in Australia?

Here are a few edits: “During a recent tennis match close to the end” can be: “Close to the end of a recent tennis match,”
Put catch up in the past: “…suddenly, he caught up with his competitor”
“Winning the part” is probably either “winning the match” or “winning the set” (in tennis, there’s a point, a game, a set, and a match.)

I’m not sure if this is different in Britain or not, but in the US, “public holiday” is to say that government and official offices are closed. To say “public holiday” is to specify that stockmarkets, banks, schools, offices, etc., are closed because it’s relevant to the sentence. For example, “The bank transfer will arrive on Tuesday because Monday is a public holiday.” This makes sense because banks are closed on Monday, so the transfer will arrive on Tuesday. However, when talking about your day to day life, simply “holiday” is enough. So I would say, “After the recent holidays, I’ve fallen behind in my goals to improve my health.”

One last thing: Intentions are just thoughts, so you can’t fall behind on an intention. You can fall behind on progress or goals.

Good example – you don’t need “on” here. You can say, “The little fish fell behind the others and were almost eaten by a shark.” There’s a funny joke in English. If you’re out in the woods with a hiking partner and you see a bear. You’re not scared. Your partner asks why you’re not scared: after all, you can’t outrun a bear. You can reply, “I don’t have to outrun the bear. I just have to outrun you.”

I think you mean “…improve the quality of their vehicles.” You can also say, “….improve the quality of their vehicles with every model year.” In the second sentence, this is a great chance to use “do so” – “…those who don’t do so can fall behind in sales.”