"Back" is used in many expressions. For example, something like "Well, it’s Monday morning. Back to the salt mines" is often said. Would the full sentence be "I'll be back the salt mines" or "I'll go back to the salt mines?" Is future tense even used in expressions like these? If I say "It failed. Back to the drawing board" am I saying that I will go back to the drawing board or am I already back to the drawing board? As another example, for an expression like "back to the playground with you," what would be the omitted verb?
Very often the missing phrase is "let's get" (the first-person plural imperative)
Well, it’s Monday morning. [Let's get] back to the salt mines
Let's get back to the drawing board.
Note that "let's" in this idiom can refer to one person or a group. It is an exhortation and encouragement, c.f. "Come on!"
The sentence "back to the playground with you" is not so obvious. Again I think it originates from an imperative.
"[Get] back to the playground with you."
This would be an inversion of the archaic imperative form, "Get you back to the playground!"
This old form of imperative can be seen in the famous song Danny Boy
But come ye back when summer's in the meadow. Or when the valley's hushed and white with snow. I'll be here in sunshine or in shadow Oh Danny Boy, oh Danny Boy, I love you so.
Today we would say, "[Get yourself] back to the playground!"
With "back," probably the best verb is "return":
Well, it's Monday morning. I have to return to the salt mines.
It failed. It has to return to the drawing board.
You, troublemaker! Return to the playground.
"Go back" works in place of return.
I've often heard and used this expression completed with "it is" or "it's" For example: "Well that failed so it's back to the drawing board."