The formula for the Bockstein $\beta:H_n(X;\mathbb{Z}/p\mathbb{Z})\to H_{n-1}(X;\mathbb{Z}/p\mathbb{Z})$ is $$\beta[c\otimes 1]=[\frac{1}{p}\partial c\otimes 1]$$ (McCleary page 456)
How about for the higher Bockstein $$\beta_r: H_n(X;\mathbb{Z}/p^r\mathbb{Z})\to H_{n-1}(X;\mathbb{Z}/p^r\mathbb{Z})$$?
I read that it is related to the connecting homomorphism, but am quite confused about the formula. Is it (just a guess based on my confused understanding):
$$\beta_r[c\otimes 1]=[\frac{1}{p^r}\partial c\otimes 1]$$
I read some texts but they didn't seem to write the explicit formula. Thanks for any help.
Any short exact sequence of abelian groups $A\xrightarrow{i}B\xrightarrow{\pi}C$ gives rise to a homological operation $i^{-1}\partial \pi^{-1}\colon H_n(X;C)\to H_{n-1}(X;A)$.
For $\mathbb Z/p\xrightarrow{\cdot p}\mathbb Z/p^2\to\mathbb Z/p$ one gets the Bockstein operation. And your second formula indeed gives an operation corresponding to $Z_{p^r}\xrightarrow{\cdot p^r} Z_{p^{2r}}\to Z_{p^r}$.