This is the 803rd most frequent Japanese word.
食い
"食い" (kui) means "eating" or "consumption."
Here, '食い' refers to the act of eating and is used in a compound verb structure with '終わる' to mean 'to finish eating'.
私は朝食を急いで食い終わりました。
I quickly finished eating breakfast.
In this context, '食い' is part of the verb '食いそびれる,' which means 'to miss eating'.
彼はギターを弾くことに夢中で、昼食を食いそびれた。
He was so engrossed in playing the guitar that he missed having lunch.
'食い' appears in the phrase '食い入るように,' which means 'intently' or 'keenly,' metaphorically derived from 'to bite into.'