For more information about, how to feed laying hens and chickens please read my article here How to keep garden hens
Please see my article for feeding laying hens looking after chickens in your garden
Yes you can feed hens garden produce and leftover cooked fruit and vegetables.
Hens housed in a coop and chicken run that don't have access to green stuff will benefit from and enjoy being fed cabbage and other fruit and vegetable kitchen leftovers.
They actually enjoy vegetable peels, bananas, apple cores, carrots, broccoli even spaghetti and porridge. In fact, Hens can be fed almost any wholesome food, vegetable or fruit you eat yourself, if you can eat it so can your hens and you can feed them this daily as part of a balanced diet that also includes the mixed corn, layers pellets and grit I have listed above.
I have found over the many years I have kept layers and birds for the table that one of the many benefits of keeping chickens is that most of my kitchen waste and stuff I grow in my garden can be fed to them ensuring my flock get a varied and balanced diet and saving me some money too.
Avoid feeding your chickens any raw green peels (such as green potato peel) and citric fruits such as oranges and lemons all fatty foods or foods with lots of salt in it, also sweets (to much sugar) and processed food (to much of everything that is bad) and poisonous garden plants and weeds like foxgloves and bindweed. If it is bad for you it is bad for your hens.
Chickens are fairly good at knowing what not to eat and will avoid anything really nasty.