Wacky shifting on my 9 speed Sora shifter. It's fine going up toward biggest cogs but skips 2-3 gears when going down from the biggest cog. It shifts fine when going down from the middle to smaller gears. Is the shifter dead or is something else at play? I've tried flushing it with wd-40 with little success.
A delay in upshifting is often caused by the cable sticking in the housing. The force trying to move the derailleur outwards comes from just the derailleur spring which sometimes cannot overcome friction between the cable and the housing. The solution is new cables and housing, or sometimes just removing and cleaning the cable will work.
However, it sounds like you are getting multiple sprocket upshifts with a single lever click with no delay. This might be a problem with the shifter ratchet mechanism. You can diagnose this by detaching the shift cable from the derailleur and pulling it out of housing sections up to the housing on the handlebar. Put tension on the cable and work the shifter. If you can get it to skip positions as you upshift the the shifter is at fault.
If you have full-length housing you can still do this procedure but you may need a friend to pull on the cable at the back if the bike as you work the shifter.