Some hydrogen atoms have neutrons. There are three different common isotopes of hydrogen. Isotopes are atoms of the same element that contain different numbers of neutrons.
Hydrogen-1 has no neutrons, just one proton. Hydrogen-2, also called deuterium, has one neutron and one proton. Hydrogen-3, also called tritium, has two neutrons and one proton. Hydrogen-3 is radioactive and undergoes beta decay.
There are a few other heavier and highly unstable isotopes of hydrogen that don't occur naturally but have been synthesized by scientists.
The most common isotope is hydrogen-1. It accounts for 99.98% of all hydrogen in nature, so it's correct to say that most hydrogen atoms have no neutrons. The number of neutrons an atom contains is influenced by a balance between the nuclear force binding protons and neutrons together and the electromagnetic force that keeps electrons around the atom. The three naturally occuring isotopes of hydrogen all fall within the range of stability for these two competing forces.