The art of improvisation and the Chaos of playing with kids
It sounds a little bit awkward, but there's indeed not an exact ruleset for improvisation. And in fact there's nothing wrong about improv, since you cannot have all the answers to all of the possible outcomes and that counts especially for young players. They just think differently.
I DMed for about 10 years in an institution for children. We played DnD, Cthulhu and my home-brew system. And you can't imagine how often I had to ask: "You want to do WHAT?"
Playing DnD with kids is like trying to ride a flea circus through a hurricane. It's pure chaos. My advice: Dump your stupid story, dump all the plans you had, let them have their fun by dismantling your world. That worked pretty fine with "my" kids at least. It's a lotta work, because sandboxing your world IS freakishly much effort. Or, just improvise or (how I did it) do a healthy mix of plan/plot and improvise.
The biggest mistake you can do is to prepare too much but also not too little that your players lose guidance; it's more like a general rule. When playing DnD with kids I went from highly prepared adventures to loosely connected event snippets ready to get discovered by the party.
How they get from one event to the next is completely improvised.
That's because kids tend to lose focus very quickly, because kids are so much more curious than adults; they will turn every stone, push any button and open every door, just for the purpose of finding out odd stuff.
In my experience there are many bad ways for a DM to react to such players. For example: Place some nasty traps just to push your players to not mindlessly stride around. Don't punish your players for having fun in your world, reward them instead for appreciating the inside of your head. They're guests there, so treat them as such.
In my experience the only real solution is to let them find odd stuff that is somehow connected to your adventure plot. Because everything they do is delivering possible plot hooks. Maybe not for your planned adventure, but maybe for something completely different.
A player wanders without focus through the woods and finds a small village that is threatened by a feral goblin tribe. BOOM, instant quest.
A player drinks from the creek and finds some strange item (ie. a message in a bottle) that gives a clear hint for your adventure.
It needs a lot of practice to get used to improv and you WILL probably fail in the beginning, but that's okay. Let your story grow and develop into something epic. They will need some time to get along, but if you just let them do what they want to do and push them smoothly in one direction they will finally get along. Most likely, they will surprise you with ideas you never thought that they'd be possible. Just present them a problem, don't plan possible solutions; just let them do their thing and you will learn completely different ways how to play DnD.
At least that was my experience that I made while DMing for kids for a decade.
A general advice for becoming more self-confident in improvisation
It's so simple that it's quite redundant: read the fluff, devour it. Be an expert of the campaign setting you're running. That works best for a home-brew setting that you soft-write. That means your world is a pretty blank slate except for the sites that are important to establish the adventure plot. Everything else is pretty much created at the spot. Just take notes if you do that and make it become canon of your world. It takes some time and a lot of sessions but during the course of your campaign it becomes more and more elaborate. I wrote a whole planet with that method and I let my players have significant influence in designing that world, by just asking them: What do you want to see/experience?
In the end it was a huge pile of meshed together genres from high fantasy to cyberpunk. But hell, my players loved it and I had a lot of fun DMing that rather improvised world. At some point the world in your mind will turn into a living one, like a chaotic anthill inside your head. At this point you can improvise almost anything. Name generators help, especially for NPC, regions and new plants (the last point gets pretty much important, if you have some kind of Alchemist in your party, because your kids will want to know all that stuff).
Playing DnD with children is harder than to DM for a party of veterans. Children don't know any of the game's conventions. A veteran party pretty much knows that "Okay, we meet at the tavern, slay that dragon, save the damsel and get rich." It's pretty straight forward. They just know the dynamics of looting and leveling. Children don't know that concept yet. That's okay, but it forces you to think differently as a DM.
Yes, from time to time it's okay to say: "Phew, I really don't know!"
But don't use it as a standard cop-out, because it's like dropping the ball. Try to not drop the ball. If a player asks you: "What happens if I drink from the creek?" go with the obvious. If you don't have another idea in mind or a reason why it's anything BUT default your answer should be: "I guess you will be less thirsty. The water is crystal clear, you can fill your waterskin in it."
I mean, why won't you know what happens if someone drinks some water?
What happens IRL if someone drinks water?
In general, if you don't know the exact outcome in your world go with what would happen IRL.
If you don't want for a single player to wander the woods, tell him that the next village is that one they're coming from and that the next one is hundreds of miles further away.
The character could know that stuff, so why not just tell the player?
Another important thing: Keep Them Busy
Give them something to do. Sometimes they won't react to obvious hints, but that's because they don't know how to read that hints yet. If there's an obvious quest giver in the middle of the market, maybe let that one talk to THEM instead of the other way around. You have to teach them how to read your world first. Take them by the hand and let them lose step by step. Kids get distracted very fast by things you describe, that also counts for any other players but with kids that is turned up to eleven.
If you describe a room for example, start with the most unimportant things first: What is the floor made of? How high is the ceiling? Are there any windows? Than describe interesting items: There's a silver sword hanging at the wall. A huge golden candelabra is standing on a wooden table next to a small metal box. And close with the most important/dangerous things: A grumpy troll is looking down to you...roll for initiative.
And that counts for a greater scope as well. Give them something to focus on; a clear goal they have to achieve, a world they can dive into. If your world is a completely blank slate, your player won't got the extra mile for you to imagine your adventure. At least you need a very clear picture of your world that you can communicate to your players. Children's imagination on the other hand is most oftenly very wild and unsteady, if you don't give them something to "chew" they will break out and "chew" on the first thing they can grab. But you can focus that raw creative power if you push them smoothly into your adventure's track by giving them a clearly communicated goal to achieve. For example: a monster kidnapped some children from the village, save the children, naturalize the monster, bring back the children, get 100gp.
I know all that stuff may seem a little bit off-topic but the less random stuff your players do, the less improv you have to do. But again look at those situations as opportunities for interesting plot hooks, that might get you of track, but that doesn't have to be a bad thing.