Dog cough came back after Temaril P. Is it allergies or something else?

Hello and welcome to Petco Pet Education Center, formerly Petcoach.
Temaril is a combination product of a steroid (suppresses the immune system, is an anti-inflammatory), as well as an anti-histamine (counters the histamine affect that is produced by allergies). Another possibility (less likely given the breed/age of Zeek) is a collapsing trachea (weak airway) where the steroid helped with the inflammation.
The fact that Zeek responded to this medication, and it has come back when off it, is suggestive that there is either an allergy that is present that requires longer treatment, or small airway disease where the anti-inflammatory effect helped open up the airways.

I recommend discussing with your vet if they feel this is seasonal allergies, and if so discusss a medication you can use longer term through the season (discuss Apoquel), as steroids long term can be hard on the body.

Best of luck with Zeek!