1:20a And as for you, my lord the king, the eyes of all Israel are upon you

My lord and king, everyone in ⌊the land of⌋ Israel is now observing/watching you.
-or-
And so all the people of Israel are waiting for your (sing) ⌊decision/instructions⌋, my royal master/husband.

1:20b to tell them who will sit on the throne of my lord the king after him.

⌊They are waiting⌋ to hear from you (sing) who will sit upon the great/royal seat to rule as king after you, my lord and king.
-or-

⌊They want⌋ you (sing) to tell them who will inherit your throne and be the next king.

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