**Core Concept**
The patient's presentation is suggestive of a possible malignancy, specifically lung cancer, given the combination of smoking history, hemoptysis, and a hard painless lump in the left supraclavicular fossa, which is a common site for metastasis from lung cancer. The clinical presentation is likely to be related to the lymphatic spread of cancer cells.

**Why the Correct Answer is Right**
The most appropriate diagnostic step in this scenario is to perform a **Fiberoptic Bronchoscopy**. This procedure allows for direct visualization of the airway, collection of tissue samples for histopathological examination, and assessment of the extent of disease spread. Bronchoscopy is particularly useful in evaluating the primary site of lung cancer and identifying potential endobronchial lesions.

**Why Each Wrong Option is Incorrect**
**Option A:** Imaging studies such as Chest X-ray or CT scan are useful for assessing the extent of disease spread and identifying metastatic lesions, but they do not provide a definitive diagnosis of lung cancer.
**Option B:** Sputum cytology is a useful diagnostic tool for lung cancer, but it may not be sensitive enough to detect cancer cells in patients with early-stage disease, especially if they are not producing sputum.
**Option C:** Fine-needle aspiration cytology (FNAC) can be used to sample lymph nodes or masses, but it may not provide a definitive diagnosis of lung cancer without histopathological examination.

**Clinical Pearl / High-Yield Fact**
The presence of a hard painless lump in the supraclavicular fossa, known as Virchow's node, is a classic sign of metastatic malignancy, particularly from lung or stomach cancer. This clinical correlation is essential for clinicians to consider in the differential diagnosis of such patients.

**Correct Answer: C. Fine-needle aspiration cytology (FNAC) of the supraclavicular lymph nodes.