Hwang1 notes that at the height of the wayward and ill-fated period of Israel’s judges, they, too, sought to combine these two items in seeking God’s guidance (Judges 17:5–6; Judges 18:1, Judges 18:5). However much they desired direction from the divine, it does not seem that they equally desired obedience. It is likely that Hosea’s combination of them here is a damning allusion to that previous period of disobedience.
As with the previous couplet in Hosea 3:4, there is an admixture here of orthodoxy and idolatry, prescribed faith and foolish paganism. The ephod was a garment worn by a priest in the practice of divination.
2 This divination involved the Urim and Thummim (Exodus 28:30). On the other hand, Hwang3 says the household gods
were an object of pagan origin, portable idols not unlike charms (Genesis 31:19), kept in homes (Judges 18:14–20) and used in divination (2 Kings 23:24; Zechariah 10:2).
So, as with Israel’s cultic life and worship, in seeking divine guidance, they blended inspired and idolatrous elements.
4 For the children of Israel shall dwell many days without king or prince, without sacrifice or pillar, without ephod or household gods.