Beseech
Vine's Expository Dictionary of NT Words [1]
the most frequent word with this meaning, lit. denotes "to call to one's side," hence, "to call to one's aid." It is used for every kind of calling to a person which is meant to produce a particular effect, hence, with various meanings, such as "comfort, exhort, desire, call for," in addition to its significance "to beseech," which has a stronger force than aiteo (see ASK). See, e.g., the RV "besought" in Mark 5:18; Acts 8:31; 19:31; 1—Corinthians 16:12 . See Call , No. 6, Note (2), Comfort, Desire, Exhort, Intreat, Pray
often translated by the verb "to beseech," in the Gospels, is elsewhere rendered "beseech," in 1—Thessalonians 4:1; 5:12; 2—Thessalonians 2:1; 2—John 1:5 . See under Ask , No. 2.
"to desire, to long for," usually representing the word "need," is sometimes translated "beseech," e.g., Luke 5:12; Acts 21:39; 2—Corinthians 10:2; Galatians 4:12 . It is used of prayer to God, in Matthew 9:38; Luke 10:2; 21:36; 22:32; Acts 4:31; 8:22,24; 10:2; Romans 1:10; 1—Thessalonians 3:10 . See Pray , Request.
Matthew 18:26
King James Dictionary [2]
BESEE'CH, pret. and pp.besought.
To entreat to supplicate to implore to ask or pray with urgency followed by a person as, "I Paul beseech you by the meekness of Christ,", 2 Corinthians 10 or by a thing as, I beseech your patience.
Webster's Dictionary [3]
(1): (v. t.) To ask or entreat with urgency; to supplicate; to implore.
(2): (n.) Solicitation; supplication.