Please see in action the famous games from the grandmasters demonstrating the combat dynamics of a Queen versus three minor pieces. No English language can explain the entire dynamics. You must see it for your self.
Generally speaking the queen is going to be better when the opponent's king is exposed to checks and/or when the position is "loose" ie pieces and pawns hanging, pawn weaknesses etc.
If the side without the queen can hold the position together and push a passed pawn there isn't really anything the queen can do.
For me I belive that the Queen is worth 3 pieces for a variety of reasons,
It combines the functionality of the Rook and the Bishop, the Bishop, which act like snipers in the early-mid game, are very valuable, because of their sniper-like capability, whereas the rook is bad in the early stages of the game, because the board is crammed usually, conversely, ...
I am presuming things such as pawns, bishops and knights count as minor pieces for what my answer will be.
Queen > 3 Minor Pieces
In the case that there is a way that you could get a checkmate, but you need to make a risk with what your opponent will notice and what he/she won't notice. You could try to put 3 pieces to get your opponent to take those ...
A lot depends on whether there are other pieces involved and the pawn structure.
If you mean positions with Q alone vs three minor pieces then relative king safety comes into play. The side with the Q tries to expose the opponents king and harass it.
The Q alone side may need to create connected passed pawns to make progress since the pieces should easily ...
The queen does a good job when there are a lot of weaknesses to attack, especially if the opponent's king is out in the open, so there are a lot of options for double attacks.
The pieces are generally stronger if they can coordinate and the king is still relatively safe. Earlier in the game that's usually the case.
With such an overwhelming pawn advantage it should be easy enough to win.
What you said is definitely a good plan. You can push h5 and g5 to push the king back.
[fen "8/R4ppp/4pk2/1r6/8/5K2/6PP/8 b - - 4 38"]
1... h5 2. Ra4 g5 3. h3 Ke5 4. Ra2 f5
We can see here that black has an easy win. g4 is coming and once the pawns are traded, Black's two ...
My plan was to keep the Black rook on the 5th or 4th rank to cutoff
the White king and advance the h and g pawns. But then what?
Your plan is a bad one.
The general rule in rook and pawn endgames is:
Your king protects your pawns
Your rook attacks your opponent's pawns
You can add another one:
Push your passed pawns
As you can see your passed pawns are ...
All black pieces are currently defended but that does not mean there is nothing to do as if you play aimlessly white will eventually gain an advantage.
On analyzing the game the first thing you can notice is that the kingside is completely locked down by pawns so our attack needs to be on the queenside. Ba6 and Be7 are good moves to start gaining control on ...
Currently e5 is not playable due to pawn loss. b3 is good as if ever e5 occurs after preparing for it with Nbd7, dxe5 dxe5 would make the long diagonal accessible to the bishop forever. In double fianchetto positions, an open centre is desired for maximum range of bishops.
Obviously, It's not strict, e5 is a very important break in many dutch positions.
Please note that the ...e5 push Black does on your line is not forced and there are plenty of other variations Black can opt for (like 4...e6, or 4...g6, or 4...Nf6 5.Nc3 d6, or even 4...e5)
Each of this alternatives requires White to be prepared as Black will probably know a lot of theory about those (and have a good understanding of the resulting ...
The main point is that 1...Nxf2 draws White's king out to the vulnerable f2-square; this allows Black to hold the balance. After 2.Kxf2 fxg4 3.Rg1, Black can give a perpetual with 3...Qh4+! 4.Ke3 (4.Rg3 can be met with 4...exf3, as Rxg7+ is impossible) 4...Qh6+ 5.Kf2 Qh4+.
Meanwhile, 1...fxg4? 2.Qxe4 gxf3 3.Rg1 (hitting the g7-pawn) 3...Rf7 4.0-0-0 gives ...