Answer: If you mow often enough, you can leave the grass clippings on the lawn where they will decompose and when doing so, will release nitrogen which will be rinsed back into the lawn with rainwater or water from the hose. The key is to mow frequently so that the clippings are only an inch or so long. If the clippings are longer, you can rake them up and spread them out so they can dry out, then use them as mulch around your perennial plants. Don't pile fresh lawn clippings around your plants. They will shed water until they decompose, which might leave your plants thirsty, even though you water. I'd recommend removing the plant debris caused by pruning or natural dieback. If you just leave the vegetation strewn all over the garden you'll provide hiding places for insects and rodents and any plant pathogens can remain alive and viable - risking the possibilty of infecting otherwise healthy plants. I would remove plant debris and pile it up in an unused part of the landscape so it can decompose, or bury it where it will decompose out of sight.
Hope this answers all your questions!
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