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This is a harmless fungus. These are present in the hay, and help to break it down into nutrients over time. This is nothing to worry about, but if you would like to get rid of it, then spray it with peroxide without getting it on the mango, which looks very healthy by the way.

Fungus in mulch is very common.
These links will help you.

https://www.gardeningknowhow.com/garden-how-to/mulch/common-mulch-fungus.htm
https://www.gardeningknowhow.com/ornamental/fungus-lichen/what-is-slime-mold-slime.htm

Yes it's a mushroom, the fruiting body (spore producing organ) of a fungus. Probably growing on the decomposing mulch. It's probably harmless but am unable to identify the type, so don't eat it unless you get a positive ID, some mushrooms are deadly poisonous. You can lift it out with a trowel or a plastic bag over your hand like dog walkers pick up poop, and throw it in the garbage. Or let it be and as you have observed in the past, they dry up and go back into the mulch and soil.

You should be able to leave the old bark mulch.

This can harm the plants, and I would advise this between crops, or before planting. White, or gray mold is normal, and harmless, and sometimes beneficial. Other colors such as dark green, black, or dark gray can indicate a problem.

I would, instead, suggest spraying with pure peroxide. Avoid the plants when this is not diluted, but spraying the mulch will be just fine.

If you have a leaf blower or can borrow/rent one, use it on the lightest setting (not holding the trigger). It will be enough air to move the leaves without moving too many wood chips.

If there aren't too many leaves and you aren't too fastidious, you can leave them to decompose. Also, I use a small width rake in my garden beds that seems easier; I also use it to get leaves out of bushes.