It's an expression most often used in a work context, but it can be extended to mean any activity. For instance, following the birth of a child, a couple may decide that one of them is "on mornings" (while the other one sleeps) and the other one "on afternoons" (while the morning person get some sleep to catch up) - meaning that morning/afternoon is when they each take responsibility for the child (theoretical example but you get the idea of the extension in meaning).

So, "on mornings" is doing some kind of activity, usually but not necessarily work, in the morning.

As in morning sickness.

Just a guess.