I am talking about something that occurred in the past, over 5-10 years.

I am talking about something that occurred in the past, over 5-10 years.

"The team may have lost , but I had still played well."

or

"The team may have had lost , but I had still played well."

or

"The team might have lost , but I had still played well."

or

"The team might have had lost , but I had still played well."

or something else?

Sorry I'm bad at this. Thanks in advance!

I'm not too sure, but I think-- "The team might have lost, but I still played well." --is correct.

Is there any way to use "may" to sound more formal? or no?

I think it's possible to interchange may and might.

When you use "had," you are saying that at a certain time in the past you already had done something.

So, when I was in high school I had already learned pre-algebra.

Oh... but I see that you are asking about may and might. May can mean the same as might.

While the team may have lost...

While the team might have lost...

While it may be true that the team lost...

Sure, the team might have lost, but...

They can both be used the same ways.

I believe the right sentence would be "may have lost". "Might" is typically used in the subjunctive case, but here you are saying that the team ALREADY lost, and "may" solves the tense conflict.