You've got the words wrong.
Upshift = go to a higher, faster, harder-to-push gear.
On the rear this means going to a SMALLER sprocket.
Unless you have a kinda rare low-normal, rapid-rise derailer (and Tiagra were never made like that) the power for the upshift comes from a spring in the derailer, pulling the derailer sideways, towards the smaller sprockets.
The shifter only feeds out cable.
On your description my first suspiscion is that the cable isn't sliding smoothly.
You can work through the whole cable run, looking for contamination, corrosion, kinks and poorly seated sheaths, but sometimes it's more time effective simply to redo the whole thing.

There are other options too, but sticky cable is the most common.