Certainly! Here is a test case generator for the given competitive programming problem "Convenience Store":

```python
import random

def tcgen():
    # Generate the map dimensions m and n
    m = random.randint(1, 20)  # Adjust the upper limit as per problem constraints
    n = random.randint(1, 20)  # Adjust the upper limit as per problem constraints
    
    # Generate the number of existing convenience stores
    num_stores = random.randint(1, min(m * n, 10))  # Ensuring at least one store and not more stores than blocks
    
    # Generate the coordinates of the existing convenience stores
    existing_stores = set()
    while len(existing_stores) < num_stores:
        x = random.randint(1, m)
        y = random.randint(1, n)
        existing_stores.add((x, y))
    existing_stores = list(existing_stores)
    
    # Generate the number of candidate locations
    num_candidates = random.randint(1, min(m * n - num_stores, 10))  # Ensuring not to exceed the available blocks
    
    # Generate the coordinates of the candidate store locations
    candidate_locations = set()
    while len(candidate_locations) < num_candidates:
        x = random.randint(1, m)
        y = random.randint(1, n)
        if (x, y) not in existing_stores:
            candidate_locations.add((x, y))
    candidate_locations = list(candidate_locations)
    
    return m, n, existing_stores, candidate_locations

# Example usage
for _ in range(5):  # Generate 5 test cases
    m, n, existing_stores, candidate_locations = tcgen()
    print(f"Map dimensions: {m}x{n}")
    print(f"Existing stores: {existing_stores}")
    print(f"Candidate locations: {candidate_locations}")
    print()
```

### Explanation

1. **Map Dimensions (m and n)**:
   - The dimensions of the city map are generated with both values ranging between 1 and 20. These bounds can be adjusted according to the problem constraints.

2. **Number of Existing Convenience Stores**:
   - A random number of stores is chosen between 1 and the minimum of \(m \times n\) and 10. This ensures that at least one store exists and does not exceed the number of blocks.

3. **Coordinates of Existing Stores**:
   - Unique coordinates are generated for the existing stores until the desired number of stores is reached.

4. **Number of Candidate Locations**:
   - A random number of candidate locations is chosen, ensuring that it does not exceed the number of available blocks after placing the existing stores.

5. **Coordinates of Candidate Stores**:
   - Unique coordinates are generated for the candidate stores, ensuring no overlap with the existing stores.

The generated test cases provide various scenarios of map sizes, existing store placements, and candidate locations, facilitating thorough testing of the solution.