noun
half-crying; on the verge of tears
Describes a state where someone is almost crying, often with a trembling voice or teary eyes, but not fully weeping. Frequently used in the phrase 半べそをかく.
叱られた子供は半べそをかいていた。
The scolded child was on the verge of tears.
彼は半べそで謝った。
He apologized, half-crying.
泣く means to actually cry, while 半べそ refers to the state just before crying or with only slight tears.
涙ぐむ means to have tears well up in one's eyes, often from emotion, while 半べそ emphasizes the verge of crying, often with a more childish or pitiful nuance.
Compound of 半 (half) and べそ (childish word for crying or tearful face). The exact origin of べそ is uncertain, but it is likely a mimetic or baby-talk word related to crying.