This is a more explicit version of Ian's answer.

From the representation theory of $\mathbb{Z}/p$, we can assume that $V=\mathbb{C}^{m+1}$ with the generator $g$ of $\mathbb{Z}/p$ acting as $g.z=(\omega_0z_0,\dotsc,\omega_mz_m)$ for some primitive $p$-th roots of unity $\omega_0,\dotsc,\omega_m$. Put $I=[0,1]$ and $W=\{re^{i\theta}:0\leq r\leq 1,\;0\leq\theta\leq 2\pi/p\}$. Then put \begin{align*} e_{2k} &= \{z\in S^{2m+1}:z_k\in I,\; z_j=0\text{ for } j>k\} \\ e_{2k+1} &= \{z\in S^{2m+1}:z_k\in W,\; z_j=0\text{ for } j>k\}. \end{align*} There is a homeomorphism $f_{2k}\colon B^{2k}=B(\mathbb{C}^k)\to e_{2k}$ given by $$ f_{2k}(z) = (z_0,\dotsc,z_{k-1},\sqrt{1-\|z\|^2},0,\dotsc,0) $$ There are continuous surjections $p_k\colon B(\mathbb{C}^k)\times[0,1]\to e_{2k+1}$ and $q_k\colon B(\mathbb{C}^k)\times[0,1]\to B(\mathbb{C}^k\oplus\mathbb{R})=B^{2k+1}$ given by \begin{align*} p_k(z,t) &= (z_0,\dotsc,z_{k-1},\sqrt{1-\|z\|^2}\,e^{2\pi it/p},0,\dotsc,0) \\ q_k(z,t) &= (z,\sqrt{1-\|z\|^2}\,(2t-1)) \end{align*} One checks that $$ p_k(z,t)=p_k(z',t') \Leftrightarrow (z=z' \wedge (t=t' \vee \|z\|=1)) \Leftrightarrow q_k(z,t)=q_k(z',t'). $$ It follows that there is a unique map $f_{2k+1}\colon B^{2k+1}\to e_{2k+1}$ with $f_{2k+1}\circ q_k=p_k$, and that this is a homeomorphism.

One can now check that the cells $\{g^ie_j:0\leq i<p,\;0\leq k\leq 2m+1\}$ give an equivariant cell structure on $S^{2m+1}$. The cellular boundary operator is $\partial(e_{2k})=\sum_ig^ie_{2k-1}$ and $\partial(e_{2k+1})=g^{u_k}e_{2k}-e_{2k}$, where $u_k$ is determined by $\omega_k^{u_k}=e^{2\pi i/p}$. In particular, in the basic case where $\omega_k=e^{2\pi i/p}$ for all $k$ we have $\partial(e_{2k+1})=g.e_{2k}-e_{2k}$.

Most of this is in Section V.5 of the 1962 book Cohomology operations by Steenrod and Epstein,

This is a more explicit version of Ian's answer.

From the representation theory of $\mathbb{Z}/p$, we can assume that $V=\mathbb{C}^{m+1}$ with the generator $g$ of $\mathbb{Z}/p$ acting as $g.z=(\omega_0z_0,\dotsc,\omega_mz_m)$ for some primitive $p$-th roots of unity $\omega_0,\dotsc,\omega_m$. Put $I=[0,1]$ and $W=\{re^{i\theta}:0\leq r\leq 1,\;0\leq\theta\leq 2\pi/p\}$. Then put \begin{align*} e_{2k} &= \{z\in S^{2m+1}:z_k\in I,\; z_j=0\text{ for } j>k\} \\ e_{2k+1} &= \{z\in S^{2m+1}:z_k\in W,\; z_j=0\text{ for } j>k\}. \end{align*} There is a homeomorphism $f_{2k}\colon B^{2k}=B(\mathbb{C}^k)\to e_{2k}$ given by $$ f_{2k}(z) = (z_0,\dotsc,z_{k-1},\sqrt{1-\|z\|^2},0,\dotsc,0) $$ There are continuous surjections $p_k\colon B(\mathbb{C}^k)\times[0,1]\to e_{2k+1}$ and $q_k\colon B(\mathbb{C}^k)\times[0,1]\to B(\mathbb{C}^k\oplus\mathbb{R})=B^{2k+1}$ given by \begin{align*} p_k(z,t) &= (z_0,\dotsc,z_{k-1},\sqrt{1-\|z\|^2}\,e^{2\pi it/p},0,\dotsc,0) \\ q_k(z,t) &= (z,\sqrt{1-\|z\|^2}\,(2t-1)) \end{align*} One checks that $$ p_k(z,t)=p_k(z',t') \Leftrightarrow (z=z' \wedge (t=t' \vee \|z\|=1)) \Leftrightarrow q_k(z,t)=q_k(z',t'). $$ It follows that there is a unique map $f_{2k+1}\colon B^{2k+1}\to e_{2k+1}$ with $f_{2k+1}\circ q_k=p_k$, and that this is a homeomorphism.

One can now check that the cells $\{g^ie_j:0\leq i<p,\;0\leq j\leq 2m+1\}$ give an equivariant cell structure on $S^{2m+1}$. The cellular boundary operator is $\partial(e_{2k})=\sum_ig^ie_{2k-1}$ and $\partial(e_{2k+1})=g^{u_k}e_{2k}-e_{2k}$, where $u_k$ is determined by $\omega_k^{u_k}=e^{2\pi i/p}$. In particular, in the basic case where $\omega_k=e^{2\pi i/p}$ for all $k$ we have $\partial(e_{2k+1})=g.e_{2k}-e_{2k}$.

Most of this is in Section V.5 of the 1962 book Cohomology operations by Steenrod and Epstein,