Am I correct in thinking that
having + past participle conveys the flowing meaning?
Tomorrow, having finished the game, I will go home.
Tomorrow, after finishing the game, I will go home.
The former is a little confusing. A proficient English speaker will still understand you, but the exact cause-and-effect relationships are unclear.
having + past participle construction is properly used in the past tense:
Yesterday, having finished the game, I went home.
This means that some time yesterday, I finished the game, and then I went home. The sentence would still make sense without the "yesterday"; it just adds some detail.
Tomorrow, having finished the game, you're applying a purely past construction to the future tense. There are constructs for this scenario, and your second sentence is one of them, but "Tomorrow, having finished" is not.
Regarding your second sentence,
Tomorrow, after I finish the game, I will go home would be a little better. It confuses readers/listeners for a moment to read/hear what the subject (I) is doing before learning who/what the subject is.
The sentences do mean the same thing. (I'd expect to hear "After I finish the game tomorrow, I'll go home," but all 3 sentences mean the same thing.)