I’m in a residential neighborhood, and I have a stray cat with kittens lodging in my backyard woodpile. 
She doesn’t seem to be comfortable around humans. Any advice on what to do? I left out a tray of water and a small amount of cat food for her today. Should I avoid petting the kittens while she’s gone away?
"Should I avoid petting the kittens while she’s gone away? "
Very much so. Might be difficult to resist the temptation but if the mother thinks you’re too much of a threat, she’s liable to recycle her kittens as food.
Or she might move them to another location which might not be as good as their current location (otherwise she’d likely have gone there instead).
A neighbor and I TNR (trap, neuter release) stray and feral cats with the help of a non-profit that allows us to do $20 spay-neuters.
If you leave them alone, the kittens will grow up feral and will produce more kittens, as will mama cat. And since she’s found a safe place AND food, she’ll come back to your woodpile in about six months to have her next litter.
If the kittens get used to humans, they can potentially be adoptable. When we have wild kittens, we put out food, especially cheap canned food, and sit close while mama and babies, when they’re ready, eat. Chances are the kittens will become quite tame if you do this regularly. Otherwise I’d suggest finding a cat rescue or TNR organization to get them trapped, speutered, and possibly adopted. Otherwise you will have kittens being born on your property for a long, long time to come. Feral cats don’t live long or well without someone providing food and minimal care.
Lots of resources on the Alley Cat Alliessite.
OK, am I the only one that read the thread title and thought it was about Julie Reiten being pregnant?
Call the local cat-homing people. They will have a procedure for dealing with stray cats and their kittens.
It probably won’t involve euthanising the mother cat or her kittens - in my area, if they can, they simply capture them, neuter them, innocculate them and release them. They will continue being feral cats but they won’t make any extra cats or spread FIV or many of the other significant cat diseases.
Sometimes charities choose to home the kittens (and occasionally the mother) if they’re able to and if the kittens are in an area where their likelihood of free-range survival is very low. Sometimes they kill them; in my area that’s just not done, but I don’t know what it’s like where you live.
If the first one you call says they’ll euthanise the cats (after you ask), then look for other alternatives.