Final answer:
The cup-like structures collecting urine at the end of each papilla are called calyces, specifically the minor calyces. These structures are integral in the passage of urine from the collecting ducts to the renal pelvis and then to the ureter.
Step-by-step explanation:
The cup-like structures that collect a client's urine at the end of each papilla are known as calyces. Specifically, these are the minor calyces, which receive urine directly from the collecting ducts. Urine then passes from the minor calyces into the major calyces, subsequently flowing into the renal pelvis, and finally through the ureter. The renal papillae are the medullary area of the renal pyramids in the kidney's medulla where the collecting ducts empty urine into the minor calyces. The collecting ducts, which are not technically part of the nephron but are continuous with it, collect filtrate from several nephrons for final modification. This system plays a crucial role in the concentration and regulation of urine.