Exegesis:

kai hupestrepsen ho Iēsous en tē dunamei tou pneumatos eis tēn Galilaian ‘and Jesus returned in the power of the Spirit to Galilee.’ This clause takes up and complements the first clause of v. 1, which see. Only in a secondary sense it says that Jesus returned from the place of his temptations. en tē dunamei tou pneumatos parallels plērēs tou pneumatos ‘full of the Spirit’ in v. 1. Both refer back to 3.22 and there is no reason to assume that the phrase means that Jesus was strengthened by the victory he had just won over the devil. The difference between plērēs tou pneumatos and en tē dunamei tou pneumatos is that the former refers to the state in which Jesus was after receiving the Spirit and that the latter brings out that the Spirit determines his actions. For the idiom cf. on 1.17 and 2.27.

kai phēmē exēlthen … peri autou ‘and news about him went out,’ or ‘news went out about him.’ The former translation takes peri autou with phēmē ‘news about him,’ the latter with the whole clause. Both are possible but the latter is preferable because of the place of peri autou at the end.

phēmē ‘report,’ ‘news’ as clearly in Mt. 9.26. The word has a rather neutral meaning and translations like ‘fame’ are not adequate.

kath’ holēs tēs perichōrou ‘throughout the whole neighbourhood,’ , where, however, the situation is different, because a specific point of activity, i.e. Capernaum, is mentioned previously. This is not the case here and this implies that the choice is between (1) Galilee and surrounding regions, and (2) the neighbourhood of Galilee, i.e. Galilee itself. The former is preferable.

Translation:

And Jesus returned … into has a continuative aspect, ‘then (or, from there) Jesus continued his return journey (or, went on) to,’ or a resultative aspect, cf. ‘then Jesus came-back in’ (Bible de Jérusalem, similarly Batak Toba).

In the power of the Spirit, or, ‘directed/guided by the power of the Spirit,’ ‘the one who was empowering him was the Holy Spirit’ (Cuyono); and cf. on “in the power of” in 1.17.

A report concerning him went out, or, ‘news about him spread’; or an equivalent expression such as, ‘people/everyone spoke about him.’

All the surrounding country, or, in receptor languages where the implied central point has to be made explicit, ‘in that region and in all those regions which were near-it’ (Pohnpeian), ‘on the-whole-of all those regions which were around-it (i.e. around Galilee)’ (Trukese).

Quoted with permission from Reiling, J. and Swellengrebel, J.L. A Handbook on the Gospel of Luke. (UBS Handbook Series). New York: UBS, 1971. For this and other handbooks for translators see here . Make sure to also consult the Handbook on the Gospel of Mark for parallel or similar verses.