This replaces a query I answered on my own. I don’t know how to just delete a query once it has appeared. 
The way I found the answer was this: I did a google search on the exact Greek word that was bothering me. Among the hits was a dictionary entry that showed me that I had confused one word with another word.
That’s my “Hail Mary” play for this kind of problem.
Why don’t you post what you found, so that others can find the answer too when they stumble across this thread? (I always have trouble keeping forms of πείθω and πάσχω apart myself).
I confused two different words,
#1. πειθω
#2. επιθuμεω.
The problematic word in the exercise was επεθυμουν, which the answer key translated “were eager to”.
I was trying to make επεθυμουν be a form of #1, πειθω, which led me into much confusion. That in turn led to my Hail Mary google search on επεθυμουν, which made me decide to stop looking at πειθω.
I’m leaving the middle/passive of πειθω for another day. I’m still too weak at punching the verb tables to get to this quickly.
“επεθυμουν”
It can be easy to miss the augment on επι- verbs. If you aren’t ready for it, you might see ε-πεθυμ-ουν (augment-root-suffix) when it’s actually επ-ε-θυμ-ουν (prefix-augment-root-suffix)
Many thanks, jeidsath!
I’m curious as to why you’d do a Google search for this before using the Perseus word study tool?
Example:
http://www.perseus.tufts.edu/hopper/morph?l=epequmoun&la=greek#lexicon
Perhaps a little morphology would have helped. There’s no way επεθυμουν could have anything to do with πείθω. πειθ- could not reduce to πεθ-, and where would the –υμ- have come from? (θυμ- on the other hands, is very common, θυμος etc.etc.) And remember that the imperfect of –ω verbs is always perfectly regular, and uses the present stem, with augment: επειθ-ον –ες –ε(ν) –ομεν –ετε –ον. Unlike Latin, where you have all those pesky conjugations to contend with.
Easy in retrospect, of course.
I haven’t figured out how to use Perseus for Greek.
There are instructions at DIY Classics for creating a keyboard shortcut to Perseus Latin or Greek here
https://diyclassics.com/2014/08/15/searching-the-perseus-word-study-tool-in-chrome-firefox/
The instructions are for the Latin dictionary, but just a tiny change makes it work for Greek, too. Hope this helps.
Many thanks donhamiltontx and mwh for the comments.