  We investigate the angular distribution of electrons that are emitted in the
ionization of hydrogen-like ions by twisted photons. Analysis is performed
based on the first-order perturbation theory and the non-relativistic
Schr\"odinger equation. Special attention is paid to the dependence of the
electron emission pattern on the impact parameter b of the ion with respect to
the centre of the twisted wave front. In order to explore such a dependence,
detailed calculations were carried out for the photoionization of the 1s ground
and 2 py excited states of neutral hydrogen atoms. Based on these calculations,
we argue that for relatively small impact parameters the electron angular
distributions may be strongly affected by altering the position of the atom
within the wave front. In contrast, if the atom is placed far from the front
centre, the emission pattern of the electrons is independent on the impact
parameter b and resembles that observed in the photoionization by plane wave
photons.
