Your question is very similar to the idea of erasure-resilient codes, which were originally introduced for RAID (redundant array of independent disks) to protect data from disk failure. They are systematic codes that are good for the type of problem you described, although most of the related research seems to focus on practical aspects and, as far as I know, not much research has been done on the theoretical side. Also, as is often the case with studies on reliability of storage, the focus of erasure-resilient codes is on data corruptions, unreadable bits, disk failure, and the like (which are all "erasures" in math) rather than bit flips. But if we forget about more practical issues and focus on math, erasures and bit flips can both be treated the same way by the notion of minimum distance, so here's some little things that are known about such codes in the math literature.
The idea is basically the same as systematic linear codes. For the sake of simplicity, we only consider the binary case here. Assume that we have a linear $[n,k,d]$ code of length $n$, dimension $k$, and minimum distance $d$. Here, the dimension $k$ and the number $n-k$ will be your $M$ and $B$ respectively. Because it's systematic, we use a parity-check matrix $H$ in standard form:
$$\begin{align*}
H &= \left[\begin{array}{cc}I & A\end{array}\right]\\
&= \left[\begin{array}{ccccccc}
1&0&\dots&0 & a_{0,0} & a_{0,1} & \dots &a_{0,k-1}\\
0&1&\dots&0 & a_{1,0} & a_{1,1} & \dots &a_{1,k-1}\\
\vdots&\vdots&\ddots&\vdots&&&\vdots&\\
0&0&\dots&1 & a_{k-1,0} & a_{k-1,1} & \dots &a_{k-1,k-1}
\end{array}\right]
\end{align*},$$
where $I$ is the $(n-k)\times(n-k)$ identity matrix and $A = (a_{i,j})$ is a $k \times k$ matrix with $a_{i,j} \in \mathbb{F}_2$. The rows of $H$ are indexed by the $n-k$ bits for "some kind of backup" in your question (or any kind of storage medium of size $B = n-k$ for that matter) and columns of $A$ are indexed by the $k$ data bits we want to protect (i.e., the original data of size $M = k$).
The backup scheme is that on the $i$th backup bit, we write the sum of the data bits according to whether $a_{i,j}$ is $0$ ("ignore") or $1$ ("add"), so that the $i$th backup bit $\beta_i$ is
$$\beta_i = \sum_{x \in \{j \ \mid\ a_{i,j} = 1\} } \delta_x \pmod{2},$$
where $\delta_j$ is the $j$th unreliable data bit we are going to protect.
It is straightforward to see that the standard syndrome decoding will detect errors on $\delta_i$ as long as the number of affected data bits are fewer than or equal to $\lfloor\frac{d-1}{2}\rfloor$; we just compare each $\beta_i$ with the sum of the corresponding data bits and see if they add up, which will give us the error syndrome.
Now, we have the assumption that the backups $\beta_i$ are more reliable than the original data $\delta_j$. In your case, all $\beta_i$ are assumed to be immune to errors.
Can we correct more than $\lfloor\frac{d-1}{2}\rfloor$ errors if we assume that $\beta_i$ are all reliable? In general, the answer should be yes. So, your question boils down to how much this assumption can increase the maximum number of tolerable errors and how we can construct systematic linear codes that take full advantage of the reliable backups.
Unfortunately, these questions appear to be widely open. Basically, we need to understand how the minimum distance changes when the $n-k$ check bits of systematic linear $[n,k,d]$ codes are chopped off to form new non-systematic linear codes of length $k$.
But it is not extremely difficult to prove that some nice combinatorial structure, when used as the $A$ part, at least roughly doubles the minimum distance compared to when $\beta_i$ can be erroneous. For instance, the following paper proved that the incidence matries of the Steiner $2$-designs forming the points and lines of affine geometry $\text{AG}(m,q)$ with $q$ odd are of this "almost doubling" kind:
M. Müller, M. Jimbo, Erasure-resilient codes from affine spaces, Discrete Applied Math., 143 (2004) 292–297.
Note that their assumption is slightly more pessimistic in the sense that $\beta_i$ are "less prone" to errors, not completely reliable like your case. So they proved something stronger than the almost doubled minimum distance.
A friend of mine proved the same thing for projective geometry, although it's not published yet. (If you're curious about the exact statement, you can find it in the language of design theory as Theorem 3.16 in our preprint http://arxiv.org/pdf/1309.5587.pdf on quantum error correction that uses the idea of erasure-resilient codes.)
The code parameters in the case of affine geometry $\text{AG}(m,q)$ with $q$ odd and $m \geq 2$ is
$$\begin{align*}
n &= q^{m-1}\frac{q^m-1}{q-1}+q^m,\\
k &= q^{m-1}\frac{q^m-1}{q-1},\\
d' &= 2q,\\
(d &= q+1\ \text{if backups are as unreliable}).
\end{align*}$$
Because the dimension $k$ is set to nearly reach $n$ as quickly as possible while having the improved distance $d' = 2(d-1)$, the distance grows slower than the length $n$ and vanishes as $n$ tends to infinity.
Restricting ourselves to the case when all column weights are uniform on the $A$ part (which happens to be a reasonable assumption for the main intended purpose of erasure-resilient codes), whether we can achieve a higher improved distance $d' > 2(d-1)$ while still keeping the same level of $k \approx n$ depends on whether the kind of extremal uniform hypergraph studied in the following recent paper can have the same high minimum distance structure as affine/projective geometry:
Z. Füredi, M. Ruszinkó, Uniform hypergraphs containing no grids, Adv. Math. 240 (2013) 302–324.
If extremal or near extremal uniform hypergraphs without the so-called grids can simultaneously avoid all "bad" substructures affine/projective geometry naturally avoids, that means we can do better.
I am not aware of results that aim for substantially improved minimum distance that grows linearly with length in the case of error correction assisted by (perfectly) reliable backups.
But more active research topics on erasure-resilient codes and related ones seem to be dominated by how to cleverly use MDS codes for given situations. For example, the following paper from USC and Facebook published this year says that Facebook and many others are transitioning to erasure coding techniques typically based on Reed-Solomon codes.
M. Sathiamoorthy et al. XORing elephants: novel erasure codes for big data, Proc. VLDB Endowment 6 (2013) 325–336.
I became interested in this type of problem as mathematics a decade ago, but the fundamental questions in a simpler and more abstract case don't seem to have gained enough attention yet.
