I started a deposit to my HSA on Dec 30, but it didn't arrive at the bank until the "next year," Jan 4. It took that long for HSA. Can I apply it to the former year?

You have until 4/15/2021 to make a contribution to an HSA for the 2020 tax year ... just confirm with them that you did indicate it was for 2020.   And next time don't ever wait until the last minute to do anything that has a deadline especially when a bank/broker is involved (not to mention the holidays)  ... you may not be as lucky next time. 

As long as the bank characterizes it as a 2020 contribution, it's ok. 

Certainly check with the HSA custodian to see that it was treated as a contribution for 2020 as Opus 17 described.  By default, custodians must treat contributions as being for the year in which the contribution is deposited unless instructed otherwise.  If they tagged it as being for 2021, the deadline for getting them to correct it to be a contribution for 2020 is April 15, 2021.