Can we run DBCC CheckDB on secondary and perform our database backups with checksum on Primary.
This will make sure if corruption happens on primary, backup will fail and Primary will use sys.dm_hadr_auto_page_repair to repair page from secondary node
Also running CheckDB on Secondary node will always keep DBCC related resource utilization on secondary only.
Best Answer
Brent Ozar has a good article regarding this here: Where to Run DBCC on Always On Availability Groups
Specifically these parts of the article are relevant to your question (his example is kind of the reverse of yours, but the same principal applies in regards to the importance of running DBCC CHECKDB on both replicas):
In short: By backing up the Primary replica without running DBCC CHECKDB on it, you could be backing up corrupt data without realizing, especially more so when you've experienced a failover event that caused your replicas to switch places in the interim, for your specific scenario.