I speak English very good!

[Martyr]

Is it something to do with the word misses and it should be "like a child would miss their blanket"? I can't figure it out if it's not.

I was under the impression that the problem was that 'their' should only be used as a plural and here it is being singular. Meaning the sentence should read "I'm going to miss you like a child misses his/her blanket.", but I personally prefer the use of their/they in this kind of situation.
That was the issue, right?

That's the issue, but I wouldn't approve of your fixes either. His/Her is too wordy, too busy, too distracting, and it takes too much attention away from more meaningful parts of the sentence.

The problem, to be clear, is that the word "child" is a singular term.
Because he's a person, we do not refer to him as an "it."
However, since we do not have a specifically stated gender, people cannot tell whether to call him a he or a she.
The common result is that people default to "they."
It seems to be a peaceful solution, but it isn't. What is happening is that people are referring to one human as many humans. Even though it is becoming more and more acceptable, there is no difference between referring to a child as "they" and referring to a car as "cars."
Everybody would get upset if you pointed out a car you liked and would always speak like, "Oh! That one is my favorite cars!"

If you haven't guessed by now, the official rule is to use the word "he" when the gender is not specified.
Unfortunately, that basic, fairly clean fix for the problem has been assaulted in recent years(not so recent anymore) by those who believe that the rule is misogynistic.
That's when the awful he/she butchery came into existence.
And now, there are actually heavy scholarly advocates of replacing the old rule of using "he" with "they," even though it is more absurd and wrong than any other idea brought up to date.

As far as I can tell, there are no problems with the original system. In theory, maybe the ungendered pronoun might someday be found to refer to a woman in some parallel universe scenarios where hypothetical situations that fleetingly come and go over the course of casual conversation take life and follow a course where, by some means unbeknownst to us, gender matters. But until that sort of discovery is made, using "he" and "his" is a lot easier than using "he/she" or using a term like "they" which is utterly wrong.

If somebody really wants to cry misogyny, then I wouldn't care if, maybe, male authors used "he" and female authors used "she."
Although that sort of change could cause some confusion, since the use of "she" has since become a tool to powerfully establish a female element to a statement. I would think that making "she" into the other generic term would cause some confusion among readers during its early stages. But that problem should probably go away quickly enough.

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Anyway, because I'm aware of all that garbage, it jars me to hear people refer to a singular person as multiple persons. I'm more upset because I know that a crooked, incorrect theory is dominating our language than I am at the sound of it, really. It sounds fine. But it isn't.