1) Does the word "hardwork" exist?
2) If exist, is the meaning same as "hard work"?
I know that usually people will write hard work, but rarely to write hardwork.
The prize recognized my hard work.
... Some people might prefer to use it in the one word form for adjectival use: 'He'd had his brilliant idea. Now it was just a matter of those typical hardwork activities of <whatever>. This usage would follow the pattern of other noun phrase like 'every day' (which forms the adjective 'everyday').
But - as my failure to think of a few credible nouns to end my example suggests - this situation would rarely (if ever) occur.
It looks like something James Joyce might have used, although, to the best of my knowledge, he didn't.