What voltage are you connecting the Huzzah board to? While a normal ESP8266 doesn't have a voltage regulator so needs to run off 3.3v, the Huzzah board has one on-board, so needs a bit more voltage to work. If you connect it to the 5V line of the Pico (the one next to GND) then it should run ok?
The ESP8266 generally runs ok off the Pico with the shim, but it's a bit tight (better with the new Pico 1v4s though). There's room on the shim for a ~47uF surface mount capacitor, and once you add that it works very reliably. If you have issues with the Huzzah you could always add a capacitor too.
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What voltage are you connecting the Huzzah board to? While a normal ESP8266 doesn't have a voltage regulator so needs to run off 3.3v, the Huzzah board has one on-board, so needs a bit more voltage to work. If you connect it to the 5V line of the Pico (the one next to GND) then it should run ok?
The ESP8266 generally runs ok off the Pico with the shim, but it's a bit tight (better with the new Pico 1v4s though). There's room on the shim for a ~47uF surface mount capacitor, and once you add that it works very reliably. If you have issues with the Huzzah you could always add a capacitor too.