"on my walk" is correct and usable in written English.
You can use it to refer to a walk you took in the past, for example: Yesterday, I discovered a new cafe on my walk.
I remember wild dogs chasing me on my walk home from Cooke School; one day, a whole pack of wild dogs camped out on our front porch, preventing me from going to school at all.
Slowing down to breathe while on my walk was not exactly a choice, but a consequence of having one lobe short of a full set of lungs.
For example, many of the businesses I observed on my walk back in 2010 were owned by people of color, even one of the three bank branches.
I set out on my walk home.
On my walk back home the truck was gone, to a purpose that remains untranslated.
On my walk I pass the park's usual kind of visitors.
Other animals I've seen on my walks are foxes and rabbits.
I used to believe that talking on a cellphone had no effect on my walking pace or directional stability.
What I saw on my walks delighted me just marginally more than what I didn't see.
Since I started offering foraging trips, the number of people I've been getting on my walks has increased and the demographic has changed too.
On my walks, I saw dozens of cardinals and yellow-bellied sapsuckers, and the occasional pileated woodpecker, hammering away for beetles.
