Urinary Tract Infection (UTI): Given the abdominal pain and the presence of a singular kidney, a UTI is a highly plausible diagnosis. The elevated eosinophils could be a secondary response to the infection. The singular kidney increases the importance of promptly diagnosing and treating any renal issues to preserve kidney function.

Nephrolithiasis (Kidney Stones): Although less common in children, the presence of a singular kidney might increase the risk due to potential concentrating defects or other underlying conditions. Abdominal pain is a hallmark symptom, and while eosinophilia is not typically associated, it could be an incidental finding.
Pyelonephritis: An infection of the kidney, which could be more severe given the child has only one kidney. The elevated eosinophils might not directly correlate but could indicate an allergic or parasitic component to the infection.
Allergic Interstitial Nephritis: This condition, often caused by medications or infections, can lead to renal dysfunction and could explain the eosinophilia. However, it's less directly linked to abdominal pain in the context of a singular kidney.

Sepsis: Although not directly indicated by the elevated eosinophils, any infection in a child with a singular kidney could rapidly progress to sepsis, a life-threatening condition. Prompt recognition and treatment are crucial.
Obstruction of the Singular Kidney: Any obstruction, such as from a stone or blood clot, could cause severe abdominal pain and must be quickly identified and relieved to preserve kidney function.
Vesicoureteral Reflux (VUR): Especially if the child has a history of UTIs, VUR could be a significant concern, potentially leading to renal scarring and dysfunction in the singular kidney.

Parasitic Infections: Certain parasitic infections, like schistosomiasis, can cause renal issues and eosinophilia. However, these are less common in many parts of the world and would depend on the child's travel or exposure history.
Eosinophilic Granulomatosis with Polyangiitis (Churg-Strauss Syndrome): A rare autoimmune condition characterized by asthma, eosinophilia, and vasculitis affecting multiple organ systems, including the kidneys. It's extremely rare in children and would be a diagnosis of exclusion.
Renal Cell Carcinoma: Although exceedingly rare in children, any abdominal mass or persistent pain in a child with a singular kidney warrants imaging to rule out malignancy.

Urinalysis and Culture: To identify UTIs or pyelonephritis.
Imaging: Ultrasound or CT scan to evaluate the kidney for stones, obstruction, or other abnormalities.
Blood Tests: Including complete blood count (CBC) to monitor eosinophil levels, electrolytes, and renal function tests.
Eosinophilic Workup: Depending on the persistence of eosinophilia, further workup might include stool tests for parasites, allergy testing, or evaluation for other systemic diseases.
Consultations: Nephrology and possibly urology consultations may be necessary for further evaluation and management, especially if the child's condition does not improve with initial treatment or if there are concerns about preserving kidney function.