Specialized cells that ingest microbes or other particles such as cellular debris?

Answer 1

Endocytosis is the action of cells engulfing other cells.

Usually when a cell tries to engulf another cell and digest it, it would need some type of enzymes such acid hydrolases or others, to break down those engulfed matter. These cells are usually referred to as Lysosome cells.

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Answer 2

Phagocytes