AlwaysOn - Running SQL Agent Jobs on Secondary read-only node writing audit information to database I am setting up an AlwaysOn Availability Group with 3 nodes. 1 Primary, 1 Secondary Read-only node and 1 secondary non-readable node.
I want to offload my maintenance jobs to my secondary read-only node which is fine. I can do this by adding a step to all my maintenance jobs to check the node. If it's the Primary node, the job will stop before moving on to the next step.

I also have a maintenance database where I record all my maintenance job information such as failures, start times, end time, durations, index/statistic names etc etc. I want to keep this database within my AG group rather than have a different maintenance DB on each node. 
So my question is.. 

IF my maintenance jobs are running on my secondary read-only node and I need this to record audit information in a maintenance database within the AG Group, will this automatically write the data to the Primary Node DB or will it try and write this to the Secondary Node database. If it tries to write to the secondary node will it fail? If it does, is there a good workaround that doesn't involve having a separate maintenanceDB on each node or keeping my maintenanceDB within it's own separate Availability Group?
No, it won't automatically write to the primary database instance. You need to connect to the primary instance and perform your data inserts on that instance (linked servers is one option to do this).

You also need to make sure you're running your jobs in the right location. For example, Transaction Log Backups can be offloaded to a secondary, but even a synchronous secondary can run behind the primary (sometimes by a lot of time). If that happens, are you going to be able to meet your RPO goals?

Index maintenance is going to have to run on the primary as it needs a writable database - you can't offload this task.

CHECKDB can be run on the secondary, but I think it should be run on all instances if at all possible. When replication happens it only transfers the log data, not data pages. If you have a disk subsystem corruption on one of your replicas that you don't run CHECKDB on then you may not find corruption until you failover.