**Core Concept**
The question is testing the knowledge of a critical investigation required to prevent long-term neurological complications in a child diagnosed with bacterial meningitis, specifically caused by Haemophilus influenzae (H. influenzae).

**Why the Correct Answer is Right**
Before discharging a child from the hospital after H. influenzae meningitis, it is essential to perform an investigation to assess the development of hearing loss, which is a common complication of bacterial meningitis. This is because H. influenzae meningitis can cause sensorineural hearing loss, leading to permanent hearing impairment if not detected early. The investigation of choice is a **high-frequency audiogram**, which is a specific type of hearing test that detects subtle changes in hearing sensitivity, particularly in the high-frequency range.

**Why Each Wrong Option is Incorrect**
**Option A:** Lumbar puncture is a diagnostic procedure used to collect cerebrospinal fluid for analysis, but it is not a discharge investigation.
**Option B:** CT scan of the head may be performed to rule out complications such as brain abscess or hydrocephalus, but it is not the primary investigation for hearing loss.
**Option C:** Blood culture is a diagnostic test used to identify the causative organism of meningitis, but it does not assess hearing loss.

**Clinical Pearl / High-Yield Fact**
It is essential to remember that hearing loss can occur in up to 10% of children with bacterial meningitis, and high-frequency audiograms are a crucial investigation to detect this complication before discharge.

**Correct Answer:** D. High-frequency audiogram.