**Core Concept**
The child's recurrent infections, particularly with multiple organisms such as S. aureus, Pseudomonas, and E. coli, suggest an underlying immunodeficiency. This condition may be due to a defect in the immune system's ability to recognize and respond to pathogens.

**Why the Correct Answer is Right**
The child's symptoms are suggestive of Hyper-IgM syndrome, a primary immunodeficiency characterized by impaired class-switching of IgM to other immunoglobulin isotypes. This leads to elevated levels of IgM and impaired responses to infections. The diagnosis of Hyper-IgM syndrome can be confirmed by measuring IgG, IgA, and IgE levels and assessing the child's ability to undergo class-switching.

**Why Each Wrong Option is Incorrect**
**Option A:** Blood culture would be useful in identifying the causative organism in the current infection but would not provide information on the underlying immunodeficiency.
**Option B:** Complete Blood Count (CBC) may show an elevated white blood cell count in response to infection but would not diagnose immunodeficiency.
**Option C:** PCR (Polymerase Chain Reaction) is a useful diagnostic tool for detecting specific pathogens but would not identify the underlying cause of recurrent infections.

**Clinical Pearl / High-Yield Fact**
Hyper-IgM syndrome is a rare primary immunodeficiency characterized by elevated IgM levels and impaired class-switching, leading to recurrent infections.

**Correct Answer:** C.