Can a Message Authenticaty Code recover original data from corrupted data?
let's say that someone has used an XOR gate with one of the inputs as some random noise, the original data would get altered to some other data set. Is such a case possible if the authentication system is poorly designed?


If you change the message $m$ than the MAC will have a different value. Let see HMAC, fips-198-1;

$$MAC(m) = HMAC(K, m) = H((K_0 \oplus opad )|| H((K_0 \oplus ipad) || m))$$ 

Let say you modified even a single bit of $m$, now we have $m'$

Concentrate on this part; $H((K_0 \oplus ipad) || m'))$, and the $H$ is an approved Hash function, means that finding a collision is negligible. We assume that you have access to the keys$.

In any case, when you modify the message $m$ to $m'$ with x-oring, you have to find a collision, i.e;
$$H((K_0 \oplus ipad) || m'))=H((K_0 \oplus ipad) || m))$$

Finding collision is very hard, and if you find you will be very famous as in SHA-1, that still require $2^{63}$ calculations.


  And if it is possible, how would the system get back the original data?


If a message is changed and there is no a MAC failure, there is no way to turn back to old value, except file history etc in your operating system.

Even there is a MAC failure, you cannot turn back to the original data. MAC's are not Error Correction Codes. 



In a better short wording from Maarten's Comment;

a MAC is there to make the message tamper evident rather than tamper proof