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 simple closed curve $L \subset M$, and let $C^k(M;L)$ denote the space of $C^k$ functions whose differential vanishes on $L$. I wonder if $C^{\infty}(M;L)$ is dense in $C^k(M;L)$? In other words, let $f \in C^k(M;L)$. Can we take an appropriate bump function $\varphi$ so that the convolution of $f$ with $\varphi$ has a differential that vanishes on $L$?
I know that in a local chart, we can pass the derivative inside the integral:\begin{equation*} dg(x) = \int \varphi(y) df(x + y) \, dy\end{equation*}for $x \in L$. But it's not clear whether/how to make this vanish by choosing a special bump function.