**Core Concept**
Painful eye movement, also known as ocular motor pain or painful extraocular muscle movements, is a symptom that can be indicative of various underlying conditions affecting the eye, orbit, or brain. This phenomenon is often associated with inflammation, infection, or mechanical irritation of the extraocular muscles or their innervating nerves.

**Why the Correct Answer is Right**
The correct answer is related to the involvement of the sixth cranial nerve (abducens nerve), which is responsible for controlling the lateral rectus muscle that abducts the eye. When the abducens nerve is affected, it can lead to painful eye movements, particularly when attempting to move the eye laterally. This is because the nerve is responsible for transmitting sensory information from the eye and surrounding structures, and inflammation or irritation of the nerve can cause pain upon movement.

**Option A:**
*Option A: Thyroid ophthalmopathy* is incorrect because, although it can cause painful eye movements, it primarily affects the extraocular muscles and surrounding tissues due to autoimmune thyroid disease. However, the pain is not typically as pronounced or debilitating as in conditions directly affecting the abducens nerve.

**Option B:**
*Option B: Cluster headache* is incorrect because, although it is a severe headache disorder that can cause eye pain, it is not specifically associated with painful eye movements. Cluster headaches are characterized by brief, extremely painful headaches that may affect one eye.

**Option C:**
*Option C: Myasthenia gravis* is incorrect because, although it is a neuromuscular disorder that can cause weakness of the extraocular muscles, it is not typically associated with painful eye movements. Myasthenia gravis is characterized by fluctuating muscle weakness that worsens with activity and improves with rest.

**Option D:**
*Option D: Orbital cellulitis* is incorrect because, although it can cause painful eye movements due to inflammation of the orbital tissues, it is not specifically associated with the abducens nerve. Orbital cellulitis is a bacterial infection of the soft tissues surrounding the eye.

**Clinical Pearl / High-Yield Fact**
When evaluating a patient with painful eye movements, it is essential to consider the possibility of an underlying neurological condition, such as a cranial nerve palsy, particularly the abducens nerve. A thorough eye examination, including assessment of extraocular muscle function and cover tests, can help identify the underlying cause.

**Correct Answer: C. Myasthenia gravis**