The action of $Diff(S^2)$ on the space of embeddings is free, but this is not true for the space of immersions. For example the Boy's surface immersion is invariant under precomposing with antipodal diffeomorphism of $S^2$. So it is not at all clear to me what the homotopy type of the space of unparametrized immersions looks like. You can get some headway by comparing with the subspace of embeddings. From this you can see that the $\mathbb{Z}/2$ factor dies. This would probably make an interesting follow up MO question.

Another example is $S^1 \to S^1$ via the double covering map. $S^1 = K(\mathbb{Z},1)$ is an infinite loop space and this is an infinite loop map, but the cofiber is $\mathbb{RP}^2$. I don't think that is even a 1-fold loop space. Do you have any examples of this happening where $n>1$, $m>0$, and the quotient map $Y \to Y/X$ is an m-fold loop map? (presumably you meant for this last condition in your question, otherwise $Y/X$ could just accidentally happen to have an m-fold loop space structure with nothing to do with $X$ or $Y$).

@ArnaudChéritat, $\Omega^2$ means the pointed douple loop space, or in Andre's notation $Maps_*(S^2,-)$. My answer was definitely about parametrized spheres. I think Andre is right about the $\mathbb{Z}/2$ coming from the diffeos of $S^2$, in which case your description in terms of rotating the sphere is also right.

@joro has found the lectures note that were at the dead link, but lectures 4 and 6 are missing. I am still wondering if there were ever any notes produced from those missing lectures?