"'ll be with the kids - likely HK, Guangzhou, and surrounding smaller towns."
That region has great food - great even for China's standard.
"are western style hotels available, with western meals?"
Yes in HK and GZ especially.
In HK you can find western food easily (but of course it would be a pity to stick to that). There are McDo's everywhere. Sigh.
In mainland China, the more reliable (I didn't say good) western food can be found in the standard international hotel chains.
"Is it possible to rent an apartment for a week or two with a kitchen, and/or to hire a cook to help with the local arrangements?"
There are serviced apartments but they do not suit your needs. How will you find a cook who speak good English and knows how to cook western food?
"Mine are 2 & 4, and really like their eggs and toast (and have food allergies too)."
It is best for you to find Chinese translation of their exact food allergy. Then you can show them to the restaurants. This is very important for any country you visit, not just China.
Not long ago I made a card for a friend that said in Chinese: "don't put MSG in my food or I will die." (He had a great trip and no MSG crises.)
Not knowing the exact "picky" nature of your children, I can't say how it will turn out. But many non-Chinese children have taken to very strange dishes in China. They like the many colors and smells and unusual texture of the dishes and seem to take a degree of "macho" delight in telling their peers later what they had eaten.
I can think of fried fish filets that non-Chinese children eagerly anticipate. Imagine, kids liking fish! Another junior crowd pleaser is - you won't believe me - jelly fish salad, which the kids gleefully call "elastic" because of the texture.
So don't write off your kids' culinary adventuresome spirit just yet.
But I agree the food allergy aspect must be taken care of any time anywhere you travel.
Bon voyage.