Learn English – How to say more cheerful in one word

You can't. Cheerfuler is not a word in English and, realistically speaking, no one would ever say that (for one, it's very difficult to pronounce). The adverb cheerfully means in a cheerful manner. It doesn't mean more cheerful. The reason why the comparative form of good is better has to do with the fact that the adjective good is not a regular adjective which means that its comparative form is not formed by preceding it with the word more (e.g. more expensive, more beautiful etc.) or by adding the suffix -er to the end of the word (e.g. faster, stronger etc.). Instead, it is formed irregularly by changing the word's morphological structure entirely: good becomes better. So, more cheerful is your only option.