Suppose we know $deg(m(x))=n-1=deg(f\_1(x))=deg(f\_2(x))$.


Suppose we know $C\_1(x),C\_2(x)$ where $deg(C\_i)=n$.


Then given $n$ values of $$C\_1(x)(x+1)m(x) +C\_1(x)(x+2)f\_1(x)\in\Bbb F\_q[x]$$ and $n$ values of $$C\_2(x)(x+2)m(x) +C\_2(x)(x+5)f\_2(x)\in\Bbb F\_q[x]$$ where $f\_1(x),f\_2(x)$ are unknown can we obtain any information about $m(x)$ so that if we have sufficiently many equations with different unknown $f\_i(x)$ and known $C\_i(x)$ we can recover $m(x)$?


$1,2,5$ are some arbitrary constants in above equations.



A polynomial recovery problem
No, you gain no information about $m(x)$ whatsoever.


In fact, suppose we know
$$ A\_{i,j} m(x\_{i,j}) + B\_{i,j} f\_i (x\_{i,j}) = y\_{i,j}$$
for $i=1,2$ and $j=1, \ldots, n.$ Suppose we know $x\_{i,j}$ and $y\_{i,j}$ as well as $A\_{i,j}$ and $B\_{i,j}$.


If $m(x)$ is any arbitrary function, we can interpolate degree $n-1$ polynomials for $f\_i$ that make these equations consistent. Therefore, your equations actually don't say anything whatsoever about $m(x)$ no matter what $n$ points you evaluate at (and no matter what $C\_i$ are doing).


