  We study the effects of non-standard interactions (NSI) and the genuine CP
phase $\delta_D$ in neutrino oscillations at low, $E_\nu \lesssim
1\,\mbox{GeV}$, and very low, $E_\nu \lesssim 0.1\,\mbox{GeV}$, energies. For
experimental setup with baseline and neutrino energy tuned to the first 1-3
oscillation maximum, we develop a simple analytic formalism to show the effects
of NSI. The vacuum mimicking and its violation as well as the use of the
separation basis play a central role in our formalism. The NSI corrections that
affect the CP phase measurement mainly come from the violation of vacuum
mimicking as well as from the corrections to the 1-3 mixing angle and
mass-squared difference. We find that the total NSI correction to the $\nu_\mu
- \nu_e$ probability $P_{\mu e}$ can reach $20\% - 30\%$ ($1 \sigma$) at
T2(H)K. Correspondingly, the correction to the CP phase can be as large as
$50^\circ$ and hence significantly deteriorates the CP sensitivity at T2(H)K.
The proposed TNT2K experiment, a combination of T2(H)K and the short baseline
experiment $\mu$Kam that uses the Super-K/Hyper-K detector at Kamioka to
measure the oscillation of the antineutrinos from muon decay at rest
($\mu$DAR), can substantially reduce the degeneracy between NSI and the genuine
CP phase $\delta_D$ to provide high CP sensitivity. The NSI correction to
$P_{\mu e}$ is only $2\%$ ($1 \sigma$) for $\mu$DAR neutrinos.
