**Core Concept**
The patient's presentation of fever and neutropenia, combined with diarrhoea after antibiotic administration, suggests an opportunistic infection caused by a gut flora-overgrown pathogen. This scenario is a classic example of antibiotic-associated diarrhea, where the disruption of normal gut flora leads to an overgrowth of a particular organism.

**Why the Correct Answer is Right**
The most likely causative agent in this scenario is Clostridioides difficile (C. difficile). The administration of amoxicillin, a broad-spectrum antibiotic, has disrupted the patient's normal gut flora, creating an environment conducive to C. difficile overgrowth. C. difficile produces toxins that cause inflammation and damage to the intestinal lining, leading to diarrhoea and potentially life-threatening complications. The toxins, known as TcdA and TcdB, work by activating the Rho GTPase pathway, leading to actin polymerization and cell rounding.

**Why Each Wrong Option is Incorrect**
**Option A:** Escherichia coli (E. coli) is a common gut commensal, but it is not typically associated with antibiotic-associated diarrhoea. While E. coli can cause diarrhoea, it is usually due to other mechanisms, such as enterotoxin production or adherence to intestinal epithelial cells.
**Option B:** Klebsiella pneumoniae is a gram-negative rod that can cause a range of infections, including urinary tract infections and pneumonia. However, it is not typically associated with antibiotic-associated diarrhoea.
**Option C:** Pseudomonas aeruginosa is a gram-negative rod that can cause opportunistic infections, particularly in immunocompromised patients. However, it is not typically associated with antibiotic-associated diarrhoea.

**Clinical Pearl / High-Yield Fact**
Remember that C. difficile infection (CDI) is a common cause of antibiotic-associated diarrhoea in immunocompromised patients, particularly those with neutropenia. The diagnosis of CDI should be suspected in patients with diarrhoea and a recent history of antibiotic use, and stool tests should be performed to confirm the presence of C. difficile toxins.

**Correct Answer:** C. Clostridioides difficile.