This is the special case annunced in the comment. We assume $M$ to be an oriented 4-manifold with the homotopy type of a 2-complex. We prove that $M$ immerses in $CP^2$.\mathbb{C}P^2$.
In case of compact 4-manifolds with boundary I have a more direct and elementary proof in my paper which does not depend on Phillips theorem.
Anyway, a bundle map $TM \to TCP^2$ T\mathbb{C}P^2$ (needed to apply the theorem of Phillips) can be constructed in this way. Firstly, endow $M$ with an almost-complex structure (which exists by our assumptions). There is a nowhere vanishing vector field on $M$, hence $TM$ splits as a Whitney sum of complex rank 1 bundles $TM = \xi \oplus \varepsilon^1$ with $\varepsilon^1$ trivial and $\xi$ a pullback of the complex universal bundle. By cellular approximation $\xi$ is a pullback of the canonical bundle $\gamma^1_1$ on $CP^1$, \mathbb{C}P^1$, so $TM$ is the pullback of $\gamma^1_1 \oplus \varepsilon^1$. Now, rank 4 real oriented vector bundles over $CP^1$ \mathbb{C}P^1$ are classified by $\Bbb Z_2$ (via the second Stiefel-Whitney class), hence $TCP^2_{|CP^1} T\mathbb{CP}^2_{|CP^1} = \gamma^1_1 \oplus \varepsilon^1$ (as real vector bundles). So $TM$ is a pullback of $TCP^2$ T\mathbb{C}P^2$ by a suitable map $M \to CP^2$ \mathbb{C}P^2$ (with values in $CP^1$).\mathbb{C}P^1$).
