How do you determine that the course does not need repeating? Is the
[Compliancy Period] null?

If so, you could use an IIf() expression and test if the field is Null. I
recommend you now wrap the expression in CVDate() to help poor ol' JET 4
understand what's meant to be returned:
   Expiry: CVDate(IIf(Compliancy Period] Is Null, DateAdd("m", [Compliancy
Period], [DateCompleted]))

You could use an Update query (Update on Query menu, in query design), and
put the expression in the Update row under the field you want to update. But
I recommend that you do not do this, unless you have a need to be able to
modify individual cases so that a person gets an extended period or a
reduced period. If you will not be doing that (and I suspect that would be
unusual), then storing the calculated value breaks one of the most basic
rules of data normalization - don't store dependent values - and you are
creating a maintenance nightmare for yourself. Why not let Access do all the
hard work instead of taking that on yourself? You can use the query with the
calculated date in any context where you could use the table, and you never
have to worry about whether you have handled every possible situation where
the data could be changed to ensure your stored calculated values are
correct.