Here's an idea which might work (I'm not sure about the technical details).

Assume $M$ is closed. Choose a self-indexing Morse function $f\: M \to \Bbb R$. This will give a handlebody structure on $M$. If the support of $c$ is contained inside the $p$-skeleton of this handlebody, then we take $U$ to be the $p$-skeleton. This will do the job in this case.

If not, we may still consider the problem of intersecting the (image of each of) the simplices of $c$ with the co-cores to the handles of index $> p$. If this intersection is transversal, then $c$ will miss these co-cores. We can then cut out small tubular neighborhoods of the co-cores, and what's left will be an open set $U$ which satisfies your condition.

In the general case, we can ask whether it's possible to make a slight perturbation the Morse function $f$ to a new self-indexing Morse function $g\: M \to \Bbb R$ in such a way that $c$ misses the co-cores of the handles of the new handlebody of index $> p$. It seems to me that some version of transversality will enable one to arrange this condition. However, this is where I don't see the details.

Here's an approach which might work (I'm not sure about the correctness of this.)

1. Assume $M$ is closed. Choose a triangulation $T$ of $M$. If the support of $c$ is contained inside the $p$-skeleton of this triangulation, then we take $U$ to be a small regular neighborhood of $p$-skeleton. This will do the job in this case.

2. If not, let's consider the dual triangulation $T^\ast$. We can ask whether or not $c$ meets the $(n-p-1)$-skeleton of $T^\ast$. If it doesn't we can let $U$ be the effect of deleting the $(n-p-1)$-skeleton of $T^\ast$ from $M$. Then it seems to me that $U$ has the correct property in this case.

3 ) More generally, we can ask whether there exists a triangulation $T$ satisfying the (2). It seems to me that it should be possible to slightly modify $T$, say by general position, so that (2) holds.

Remark: I originally conceived of a version of this using Morse theory, but then realized that I had to retract it because I got confused. Perhaps it's possible to find a Morse function $f\: M \to \Bbb R$ such that $c$ misses the $(n-p-1)$-skeleton of the handlebody defined by $-f$? If so, we can define $U$ to be $M$ with this skeleton removed.

Here's an idea which might work (I'm not sure about the technical details).

Assume $M$ is closed. Choose a self-indexing Morse function $f\: M \to \Bbb R$. This will give a handlebody structure on $M$. If the support of $c$ is contained inside the $p$ skeleton of this handlebody, then we take $U$ to be the $p$-skeleton. This will do the job in this case.

If not, we may still consider the problem of intersecting the (image of each of) the simplices of $c$ with the co-cores to the handles of index $> p$. If this intersection is transversal, then $c$ will miss these co-cores. We can then cut out small tubular neighborhoods of the co-cores, and what's left will be an open set $U$ which satisfies your condition.

In the general case, we can ask whether it's possible to make a slight perturbation the Morse function $f$ to a new self-indexing Morse function $g\: M \to \Bbb R$ in such a way that $c$ misses the co-cores of the handles of the new handlebody of index $> p$. It seems to me that some version of transversality will enable one to arrange this condition. However, this is where I don't see the details.

Here's an idea which might work (I'm not sure about the technical details).

Assume $M$ is closed. Choose a self-indexing Morse function $f\: M \to \Bbb R$. This will give a handlebody structure on $M$. If the support of $c$ is contained inside the $p$-skeleton of this handlebody, then we take $U$ to be the $p$-skeleton. This will do the job in this case.

If not, we may still consider the problem of intersecting the (image of each of) the simplices of $c$ with the co-cores to the handles of index $> p$. If this intersection is transversal, then $c$ will miss these co-cores. We can then cut out small tubular neighborhoods of the co-cores, and what's left will be an open set $U$ which satisfies your condition.

In the general case, we can ask whether it's possible to make a slight perturbation the Morse function $f$ to a new self-indexing Morse function $g\: M \to \Bbb R$ in such a way that $c$ misses the co-cores of the handles of the new handlebody of index $> p$. It seems to me that some version of transversality will enable one to arrange this condition. However, this is where I don't see the details.