When seeking to extract understanding of a word it is always important to check a relevant lexicon. Etymology is often misleading. And of course context is paramount. Here is the entry from BDAG:
ὑπωπιάζω (on the v.l. ὑποπιάζειν s. W-S. §5, 19 note, end; Mlt-H. 75) (‘strike under the eye, give a black eye to’ Aristot., Rhet. 3, 11, 15, 1413a, 20; TestSol 2:4 D [ὑποπ.]; Plut., Mor. 921f; Diog. L. 6, 89)
① to blacken an eye, give a black eye, strike in the face lit. τινά someone, of a woman who is driven to desperation and who the judge in the story thinks might in the end express herself physically ἵνα μὴ εἰς τέλος ἐρχομένη ὑπωπιάζῃ με so that she might not finally come and blacken my eye Lk 18:5. Hyperbole is stock-in-trade of popular storytelling. Some prefer to understand ὑπ. in this pass. in sense
② to bring someone to submission by constant annoyance, wear down, fig. ext. of 1 (s. L-S-J-M s.v. II, NRSV, REB, et al.). In this interp. ὑπ. in Lk 18:5 has its meaning determined by εἰς τέλος. But in such case the denouement lacks punch, for the judge has already been worn down and wants nothing added to the κόπος that he has already endured. A more appropriate rendering for a fig. sense would be browbeat.—JDerrett, NTS 18, ’71/72, 178–91 (esp. 189–91): a fig. expr. (common throughout Asia), blacken my face = slander, besmirch underlies ὑπ. here.
③ to put under strict discipline, punish, treat roughly, torment, also fig. (cp. Aristoph., Fgm. 541 πόλεις ὑπωπιασμέναι) 1 Cor 9:27 (of the apostle’s self-imposed discipline. But the expr. is obviously taken fr. the language of prize-fighting vs. 26; on the virtue of self-control cp. X., Mem. 2, 1, 1; 5).—DELG s.v. ὄπωπα E. M-M. TW.
Arndt, W., Danker, F. W., Bauer, W., & Gingrich, F. W. (2000). A Greek-English lexicon of the New Testament and other early Christian literature (3rd ed., p. 1043). Chicago: University of Chicago Press.
You will notice that the lexicon makes no suggestion that what is intended is actual physical abuse (as some commenting have expressed concern about) but rather that his strict discipline and the abuse to which he allows himself to be subjected (such as in being persecuted by whipping and stoning) is "a beating".
Some think Paul had an eye affliction based on:
NIV Gal 4:15 Where, then, is your blessing of me now? I can testify
that, if you could have done so, you would have torn out your eyes and
given them to me.
But there is no suggestion that he damaged his own eyes in severe self-affliction.