  We study effects of the next-next-nearest-neighbour antiferromagnetic ($J_3 <
0$) interaction on critical properties (or phase diagram) of the frustrated
spin-$\frac{1}{2}$ $J_1-J_2-J_3$ Ising antiferromagnet on the honeycomb lattice
by using the effective-field theory with correlations. Beside the ground-state
energy, we find that there is a region of $J_3 < 0$ in which the frustrated
honeycomb lattice antiferromagnet exhibits a tricritical point, at which the
phase transition changes from the second order to the first one on the line
between N\'eel antiferromagnetic and paramagnetic phases.
