 Solution

Take the indicator function $\theta (x)=1_{\{x\ge k\}}$. Its expectation under the standard Cauchy distribution is the desired upper-tail probability. To use the proposed tail probability density function in the remaining parts, assume $k>0$: a probability density function proportional to $x^{-2}$ on $[k,\infty )$ is not normalizable when that interval contains zero.