To me, the audio sounds like "ay-PAN-kay-ree-ma" or "επάνκερημα", which is the answer I gave, even though I have no idea whether there's such a word in Greek. (Google Translate says it means "coat," but the translation doesn't work in reverse, so I'll stick with παλτό.) Anyway, I wrote what I heard, as instructed.
But it definitely sounds like there's an extra "ee" sound in the penultimate syllable. I reported that the audio does not sound correct on 8 February 2020.
Also, while I'm at it here, I wish that Duolingo would periodically review the sounds made when two consonants are combined, like γγ = ng, μπ = b, etc. As a native English speaker, this does not come easily.
(Not that I'm any stranger to all the many irregular spellings and pronunciations of English! It would just be helpful to have a periodic review in Greek -- intended as constructive feedback for the course developers -- thank you!)
"επάγγελμα" sounds correct to me, but then I'm familiar with it, "ay-PAN-kal-ma". But the "el" is quite clear. They can also be found in the Tips & notes of the first units found under TIPS at the start of each lesson.
As for reviews, if you have seen these from the Links we give:
This has a full explanation of all the alphabet as well as double consonants etc so you can review as often as you want.
I also hear the "ee" sound, and I hear it in google translate as well. But I think it is an artifact of articulation, not an actual anaptyxis. I can more or less reproduce it by changing the way I form the 'L' consonant. In my native English I use only the tip of my tongue against the roof of my mouth right behind my upper teeth to enunciate "L". This makes what I would describe as a "hard L" sound. The preceding double g ("γγ") in "-άνγγελμα", however, positions my tongue at the back of my throat and encourages me to make a "soft L" sound by putting the middle of my tongue against the roof of my mouth further from my teeth. When I do this relaxedly, I produce a similar 'ee' sound to what I am hearing.
Apparently επάγγελμα it's a cognate with profession, and since it appears to be a synonym with "vocation" and "calling" it may be used in the sense you're asking, but I'm not a greek native speaker, so I don't know for sure :)