Bring a large pot of salted water to a boil. This will cook the cavatelli perfectly al dente.
In a large skillet, heat 2 teaspoons of olive oil over medium heat. Add the garlic and cook until fragrant and golden, about 1 minute.
Stir in the tomatoes and 1/4 teaspoon salt. Let them soften for about 3 minutes, releasing their juices.
Add the corn and zucchini, cooking for 2–3 minutes until tender and crisp.
Pour in the marinara sauce, seasoning with salt and black pepper as needed. Heat through for 1 minute.
Meanwhile, cook the cavatelli according to its package instructions. Reserve some pasta water before draining.
Toss the cooked cavatelli with the marinara-vegetable mixture. Stir in the Pecorino Romano, the remaining olive oil, and adjust seasoning with salt and black pepper to taste. Thin the sauce with reserved pasta water if needed.
Serve hot, garnishing with fresh basil and additional Pecorino Romano as desired.