Modification of a CW-Complex $X$ with control over Homology groups

Question

Let $X$ be a finite dimensional CW-complex, ie we endow it with CW structure such that there exist a $n_0>0$ such that there are no $n$-simplices for $n>n_0$. Let $[\beta] \in H_k(X, \mathbb{Z}), k \le n_0$ a nonzero homology class.

Question: Is it possible to modify $X$ to another CW-complex $X'$ in following good controlled manner:

1. There exist cellular map $f: X \to X'$, ie commutes the differentials of associated complexes $(C(X), \partial)$ and $(C(X'), \partial')$

2. the induced maps $f_i: H_i(X) \to H_i(X')$ are all isomorphic for $i \neq k$

3. $f_k: H_k(X) \to H_k(X')$ is surjective and $f_k([\beta])=0$

Is such modification always possible and if yes is there any geometrical intuition behind this construction?

Motivation: For homotopy groups we can always 'kill' a class $[\alpha] \in \pi_k(X)$ by modyfying $X$ via glueing a $(k+1)$-ball along map $\alpha: S^k \to X$ representing the class $[\alpha]$ and obtaining modified space $X':=X \cup_{\alpha}D^{k+1}$ which does the job for all homotopy groups $m < k$. On the other hand this modification not doesn't give any control over homotogy groups with $m >n$. The general slogan is that 'homology is easier to control that homotopy', so I would like to know if it's possible to modify $X$ to a space $X'$ satisfying properties 1,2 and 3.

My ideas: The chain complex $(C_n, \partial_n)_{n \in \mathbb{N}}$ comprises by construction in each degree $n$ of the free $\mathbb{Z}$-module $C_n$ generated by $n$-simplices and differentials $\partial_n: C_n \to C_{n-1}$. If we want to kill cycle $[\beta]\in H_k(X)$ say represented by a chain $\beta= \sum_i c_i \sigma_i^k$ with $\sigma_i^k$ the $k$-simplices and only finitely many $a_i \neq 0$, we have to add some $k+1$-simplices to $C_{k+1}$, making $\beta$ a boundary, ie an image under $\partial_{k+1}$. Is it always possible?

If yes, there is another new problem. We want $H_{k+1}(X)=H_{k+1}(X')$, so these new $k+1$-simplices cannot contribute to homology, so must be boundaries too, so we play the same game with $k+2$-simplices. But I not see how can finish this adding new simplicies game when arriving $H_{n_0}(X)$.