**Core Concept**
Hyperuricosuria and gouty arthritis in a child with neurodevelopmental delays, self-mutilating behavior, and reduced urinary output are indicative of a disorder of purine metabolism. The enzyme deficiency in question is crucial for the breakdown of purines, leading to their accumulation in the body.

**Why the Correct Answer is Right**
The correct answer is HGPase (Hypoxanthine-Guanine Phosphoribosyltransferase), an enzyme involved in the purine salvage pathway. HGPase deficiency leads to an inability to recycle purines, resulting in their accumulation and subsequent excretion in the urine. This accumulation of uric acid can cause gouty arthritis, kidney stones, and renal failure. The neurodevelopmental delays and self-mutilating behavior are characteristic of Lesch-Nyhan syndrome, a genetic disorder caused by HGPase deficiency.

**Why Each Wrong Option is Incorrect**
**Option A:** HGPase deficiency is the correct answer, not HGPase overactivity or increased activity, which is not associated with the symptoms described.
**Option B:** Adenosine Deaminase (ADA) deficiency is associated with severe combined immunodeficiency (SCID), characterized by recurrent infections and failure of lymphoid cell development, not gout or neurodevelopmental delays.
**Option C:** Acid Phosphatase (APase) is an enzyme involved in the hydrolysis of phosphate groups from proteins, nucleic acids, and other molecules, but its deficiency is not associated with the symptoms described.
**Option D:** Acid Maltase deficiency is a glycogen storage disorder characterized by muscle weakness and hypotonia, not gout or neurodevelopmental delays.

**Clinical Pearl / High-Yield Fact**
Lesch-Nyhan syndrome is a classic example of a genetic disorder caused by a deficiency of a single enzyme, highlighting the importance of purine metabolism in maintaining normal cellular function.

**Correct Answer Line**
✓ Correct Answer: A. HGPase