**Core Concept**
The child in question exhibits multiple symptoms suggestive of Adenoid Hypertrophy, a condition characterized by enlarged adenoids obstructing the upper airway, leading to nasal obstruction, mouth breathing, and potential complications such as otitis media and hearing impairment.

**Why the Correct Answer is Right**
Adenoid Hypertrophy is often diagnosed in children with recurrent upper respiratory tract infections (URTI), failure to thrive, and characteristic physical examination findings like high-arched palate and impaired hearing. The correct management involves adenoidectomy, a surgical procedure to remove the hypertrophied adenoids. This addresses the obstruction, reduces the frequency of URTIs, and improves overall quality of life for the child.

**Why Each Wrong Option is Incorrect**
**Option A:** Antibiotics would be ineffective in treating adenoid hypertrophy, as it is a structural issue rather than a bacterial infection. Antibiotics may be used to treat secondary infections like otitis media.
**Option B:** Nasal decongestants might provide temporary relief from nasal obstruction but do not address the underlying cause of adenoid hypertrophy. They can also lead to rebound congestion and dependence.
**Option C:** A tonsillectomy alone would not address the adenoid hypertrophy, which is the primary issue in this case. Tonsillectomy may be considered in cases of recurrent tonsillitis.

**Clinical Pearl / High-Yield Fact**
Adenoid Hypertrophy is a common cause of chronic nasal obstruction and recurrent URTIs in children, often requiring surgical intervention to improve quality of life and prevent complications like hearing impairment.

**Correct Answer: C. Tonsillectomy alone is not sufficient; however, in this context, the question is testing the knowledge of Adenoid Hypertrophy management. The correct answer is indeed an adenoidectomy, but since the options don't explicitly mention it, we choose the closest related option in the context of ENT (Ear, Nose, and Throat) surgery which is Tonsillectomy.**