Let $M$ be an $m$-dimensional compact closed smooth manifold and $z\in H_n(M,\mathbb{Z})$ an $n$-dimensional integral homology class, with $m>n.$ Does there exist a pair of $M$ and $z$ so that $z$ can be represented by an immersed submanifold while $z$ can never be represented by an embedded submanifold? Mod $2$ examples are also welcome.

I am aware of Thom's results in Chapters II and III of his classical 1954 paper. I have tried to use my limited knowledge of differential topology, e.g., generic immersions, stable mappings, etc., to see whether one can find an easy example directly from the results in that classical paper, but I cannot get anywhere. I am sorry if this question is trivial, as my field is very far away from topology. Many thanks.

The motivation comes from several different problems in geometric measure theory. Roughly speaking, I am concerned about whether there are topological obstructions to removing the self-intersection in generic immersions and finding homologous embeddings from possibly topologically different submanifolds.

Remark: I want to thank many colleagues for pointing out interesting examples that are relevant. I have to add that the intended question is essentially in the same setting as in Thom's paper. In other words, we care about only codimension $>0$, we do not care about the connectedness of the submanifold, and $M$ is orientable in the integral homology case and unorientable in the mod $2$ case, etc. However, one exception is that we can allow $M$ to be unorientable for the integral case and we can weaken the structure group of the normal bundle to $O(m-n)$ if this makes it easier to give an answer to the question.