It’s either a typo or the author wants the reader to read the word “meant” the way their character would say it. When I read “mean’t” I’m imagining someone with a southern drawl. If that isn’t consistent with the character then it’s probably a typo. “Mean’t” is not a proper construction.

You know how important the singing competition was to me.

Meant is not a contraction. It is the past tense of mean.

Agree with everyone. It is an error. "Meant" is the past tense of "mean". So, if speaking in present tense: "You know how badly that singing competition means to me," hope that helps.

"you know how much that singing competition meant to me"