Wait. What? You need two 120V circuits at 20A? At 240V, that's only 20A. You don't double both of them.

Your small loads are your enemy

US Mains power has two opposing 120V "hot" wires on either side of a neutral wire. All of them suffer voltage drop in proportion to amps actually flowed (which is what we measure). In a perfect world, you have two 120V loads almost equal. In that case power comes down hot1, passes through appliance 1 via neutral to appliance 2 and returns on hot2. You have voltage drop, but it occurs off of 240V, so it's not so bad.

However, if you only have 1 large load plugged in, you have a problem. Current comes from hot1, and returns on neutral, and the voltage drop hits you harder because it's a percentage of 120V instead of 240V. This necessitates stupid-large wires.

I thought of using a multi-wire branch circuit and quickly realized the above is going to be a deal-killer. Then I thought "A subpanel, then", but a subpanel doesn't fix it either. A transformer does. If a transformer is involved, it will only pull off the 240V supply, and thus will distribute any 120V load evenly for the lower voltage drop. Needless to say I prefer a transformer based solution especially since your power needs are right-sized for the 5KVA transformers often seen on Craigslist for $100ish. However, there is also a 2-transformer solution that will greatly cut your wire costs (or alternately, give you a solid expansion path on existing wire).

Multi-wire branch circuit

In this you run a single "super circuit" consisting of 2 hots, 1 neutral and 1 ground wire. You do not need a subpanel or ground rod by Code, but a ground rod is a good idea, as ground differentials could be dangerous.

You wire one set of receptacles to hot1 and neutral. You wire the other set of receptacles to hot2 and neutral.

Design for the load you expect to be normal, not the breaker rating, and spec voltage drop for that. At the very least you shouldn't plan to load a breaker beyond 80%, so 16A is the largest number you should use. Given the distance I'd be happy with 5-6% voltage drop. If it starts proposing #8 or larger copper wire, switch to aluminum and price both ways.

What do you put for voltage? Because of the problem mentioned above, you really need to specify 120V, and that's where you get murdered. 16A/120V/400'/6% = 4AWG Aluminum @ 4.88%. Jiminy Cricket!

Mind you, if you had two large loads going, that would halve the voltage drop to 2.44%, because the drop would be applying to 240V not 120V.

In a Multi-Wire Branch Circuit you are breakering that for 20A because 20A sockets require a 20A breaker. That's all you'll get out of that. Ouch.

Subpanel

With only one large load, all the problematic math above still exists. A subpanel is not a magic wand to fix that.

However, a subpanel will have you put a 20A breaker on each receptacle circuit. That complies with the breaker requirement. And that means the feeder breaker can be bumped to the wire capacity of 60A, because feeder breaker ampacity doesn't care about voltage drop. That means you can put more circuits in it. For instance again, assuming 4AWG Aluminum, if you add a 31A/240V kiln (using a 40A breaker), you'll get 4.72% drop. That with a 12A/120V load running will cause 5.7% drop to the kiln and 4.2% drop to the blower. I could live with that.

One Transformer

OK. Let's say you fit a transformer at the gazebo. Since you want two 20A@120V circuits, that being 4800W or 4.8 KVA, a 5KVA transformer will suffice. It takes a 240V-only feed (you only need 3 wires; the wire savings helps) and provides 240V/120V to the subpanel-- hold on, that'd be a MAIN panel since it's transformer-fed, so you tie neutral and ground.

Now, your worst-case 16A/120V load gets turned into an 8A/240V load, and #10Cu feeder (though it's about the same cost as #6Al, so why are we bothering?) would suffice for 2.8% drop. However if you are heavily loading both 20A circuits, it's a wash - 16A each = 5.6% drop with #10Cu, unless we bumped to 8Cu or 6Al, for 3.68% drop. And this won't leave you any headroom for anything else.

If you want to add a bigger 240V-only load later, don't feed it through the transformer. Have the house supply come to an additional 240V-only - well, it'll be a subpanel since it's directly wired from the house. That subpanel then serves your new big loads, and also the transformer that's right there. The transformer carries on making 120V power for the 120V loads.

TWO transformers

We use two 120/240-240/480 transformers of the 5KVA variety ($100ish each) with the primaries jumpered to 480, and back to back. All this should be fully enclosed and locked down so nobody can get to the primary side of the transformers. At the house, we jumper the transformer secondary for 240V, and feed it with a 240V/20A breaker. (4800W ~= limit of a 5000VA transformer). At the gazebo, we connect that transformer's secondary same as above, to a subpanel.

Now, a single 16A@120V load is transformed to 4A@480V. Heavy use of both circuits, 16A@120V*2, loads it to 8A@480V. Let's run that through the voltage drop calc: 400'/480V/8A/5% => 14AWG @ 3.57% or 2.31% @ 12 AWG. That makes for rather cheap wire!

Also, we only need 2 wires - the 480V run is isolated from both ends, so there's no need to run a ground wire.

Did you catch the part where we just used two #14 wires to carry 3840W 400'.

Now, let's think toward expansion. Let's say instead, we get two 15KVA transformers (pricey) and do the same thing, aiming to deliver 60A@240V to the gazebo. How will that punch down? 60A breaker means realistically 48A max load, so 48A@240V becomes 24A@480V. Voltage calc sez.... 4.20% @ 10AWG, or 2.76% @ 8Cu or 6Al.

ditto ditto 25 KVA transformer and 100A (practically 80A) - 40A@480V = 4.60%@ 8Cu/6Al, or 2.95%@ 6Cu/4Al.

So... 6-4 AWG Al is a solid choice that will last through almost anything you'd plan to do with it, and if you need to upgrade, you can just throw transformers at it instead of pulling wire.