Let $ M $ be a compact smooth manifold. For $ k \geq 1 $, let $ C^k(M) $ denote the space of real-valued functions on $ M $ of class $ C^k $, equipped with the uniform $ C^k $ norm—that is, the sum of the sup norms of the function and its first $ k $ derivatives, defined using an auxiliary Riemannian metric. It is well-known that $ C^{\infty}(M) $ is a dense subset in $ C^k(M) $ under this topology.
Now, I am considering a smaller subset of $ C^k(M) $. Fix a finite subset $ A $ of $ M $, and let $ C^k(M;A) $ denote the space $C^k$ functions whose differential vanish at all points of $ A $ (for instance, we can take all non-negative functions on $M$ that vanish on $A$). I wonder if $ C^{\infty}(M;A) $ is dense in $ C^k(M;A) $? In other words, can we perturb the smooth approximation of an element $ f \in C^k(M;A) $ so that it satisfies the same constraints as $ f $? I guess we need to choose some deliberate bump functions but don't know how to carry out the details.