sich hingeben
to devote oneself
To dedicate oneself completely to an activity, feeling, or passion and become absorbed in it.
Nach einem langen Tag gab er sich ganz der Musik hin.
After a long day, he devoted himself completely to music.

to devote / hand over / sacrifice
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to devote oneself
To dedicate oneself completely to an activity, feeling, or passion and become absorbed in it.
Nach einem langen Tag gab er sich ganz der Musik hin.
After a long day, he devoted himself completely to music.

'Hingeben' (to devote/hand over/sacrifice) is a separable verb. The prefix 'hin-' stands at the end in main clauses. In subordinate clauses and in the infinitive with 'zu', the verb stays together.
Er gibt sich ganz der Malerei hin.
He devotes himself completely to painting.
In the meaning 'sich widmen' (to devote oneself), the verb is reflexive, and the thing one devotes oneself to is in the dative. The structure is: sich (accusative) einer Sache (dative) hingeben.
Am Wochenende gibt sie sich ganz ihrer Leidenschaft für das Gärtnern hin.
On the weekend, she devotes herself completely to her passion for gardening.
When 'hingeben' means 'überreichen' (to hand over), it usually needs a person in the dative (to whom one gives it) and a thing in the accusative (what one gives).
Er musste dem Beamten seinen Pass hingeben.
He had to hand over his passport to the official.
In the meaning 'opfern' (to sacrifice), 'hingeben' (to sacrifice) is often used with very valuable, abstract things. Typical objects are 'sein Leben' (one's life), 'seine Jugend' (one's youth), or 'sein Vermögen' (one's fortune).
Die Heldin im Film war bereit, ihr Leben für ihre Freunde hinzugeben.
The heroine in the film was prepared to sacrifice her life for her friends.
to hand over / to vomit
'übergeben' and 'hingeben' can both mean that you give something out of your hand. However, 'übergeben' usually sounds more formal or official, for example with documents, keys, or an object during an inspection. 'hingeben' in this meaning is rarer and often sounds elevated or literary. In everyday language, people would usually say 'geben' or 'überreichen' instead. Learners may confuse the words because both have to do with giving. A practical tip: for official or clear handovers, 'übergeben' sounds much more natural.

Der alte König gab dem Boten den Ring hin und bat ihn um Hilfe.
The old king handed the ring to the messenger and asked him for help.
Nach der Besichtigung übergab der Vermieter der neuen Mieterin die Schlüssel.
After the viewing, the landlord handed over the keys to the new tenant.