We'll consider only the case $C_p(A)$ ( and so $A$) nonsingular.
The $\binom{n}{p}$ minors of size $p\times p$ obtained from $p$ rows of the matrix determine the span of those $p$ rows ( Plucker coordinates). Hence we can determine the span of any $p$ rows, and by intersection, any row up to proportionality. Similarly for columns. So now we know the rows of $A$ up to proportionality, and also the columns. That is we have
$$a_{ij} = \lambda_i \cdot b_{ij} \\ a_{ij} = \mu_j \cdot c_{ij}$$
with $(b_{ij})$, $(c_{ij})$ known. To find $(a_{ij})$ (up to proportionality):
$$\frac{b_{ij}}{c_{ij}} = \frac{\mu_j}{\lambda_i}$$ that is, a matrix of rank $1$. So find $\mu_j$, $\lambda_i$ up to proportionality, and hence $(a_{ij})$.
Conclusion: $A$ is determined up to proportionality ( a $p$-th root of $1$).
Notes:
In the generic case, $A$ does not need to be square.
The case $p=n-1$ is simple since $C_{n-1} ( C_{n-1} (A)) \simeq A$ ( proportional)
One can recover powers of minors larger than $q$ by using the formula for the determinant of a compound matrix.
Using condensation formulas one can recover a fixed Hadamard power of $A$ ( up to proportionality).