(Top edit: The Question asserts "Xenon and krypton are popular despite their heavy mass" and asks about exploring H or He ion propellants for improved Isp. This answer shows that lighter is not better for ion thrusters, because Isp is not the proper measure of a power-limited situation. Hence, although lighter atoms have been explored for other reasons, they're certainly not explored because they provide better Isp.)
Typical ion thrusters have a small mass of propellant compared to the mass of the power generation system plus the rest of the spacecraft. In that case, the goal is to get as much thrust from the ions as possible given the power that's available.
Referring to the first equation from the Question:
$$p = \sqrt{2mqV} = \sqrt{2mE}$$
for a fixed amount of energy E, the greatest outgoing momentum hence greatest thrust comes from a larger mass atom. Switching from H to Xe is about a $\sqrt{131} \approx 12$ times increase in thrust, at the price of adding a couple kilograms to a much more massive spacecraft.
It's true that a heavier atom is ejected slower, as $E =1/2 m v^2$ means $v = \sqrt{2E/m}$. But that's more than made up for but the larger $m$ in $mv$.
Dawn is beyond the small-thruster regime into the ion engine region. It launched with 425kg of Xe on a 750kg spacecraft.
The Dawn spacecraft carried 425 kilograms (937 pounds) of xenon
propellant at launch. Xenon was chosen because it is chemically inert,
easily stored in a compact form, and the atoms are relatively heavy so
they provide a relatively large thrust compared to other candidate
propellants.
(Quote on this Dawn page)
The same number of H atoms would be only about $425/130 = 3.3 \rm{kg}$. But with the power available, the thrust would go down by a factor of 12 (although acceleration drops a bit less, as average total mass has gone down by about a sixth). That would have adverse impact on the mission. And the only way to restore the original thrust hence acceleration with H fuel would be to increase the size of the power provided by a similar factor of about 12. Dawn's solar arrays (which power the entire craft, not just the engines) are $18\rm{m}^2$ now; you'd be adding another $100\rm{m}^2$ or more, with consequent increase in mass, need for more thrust, etc.
In discussion, it's argued that what matters is the velocity of the exhaust, not the momentum. This is only true in a specific approximation where the energy of the outgoing exhaust is not intrinsically limited by some other process. For example, if you're combusting 10kg of LOX LH2, then you want that mass to be ejected with as much speed as possible using as much of the combustion energy as possible. For a constant mass (flow), it is speed that matters. But ion propulsion is (so far? usually?) limited by available power, which is a different regime. You can't compare two different mass flows without taking into account how much the available power can accelerate them.
So how does the power limit come in? Here, a higher velocity of the charged particles in the exhaust works against you. The current is $qv$, so the power needed is $qvV$: Higher velocity is more energy needed per unit of charge. Since you're limited by the energy you can put in to the exhaust stream, the exhaust velocity is effectively fixed for the
thruster.
Analytically, the available power is given by voltage and current (capital letters are electric quantities, lower case are mechanical, the $i$ subscript is per-ion):
$$ P = I V$$
Break down current into total charge per second and velocity:
$$ P/V = I = q_i dN_i/dt v$$
where $dN_i/dt$ is the number of ions exhausted per second. Expressing this in terms of the ion's intrinsic charge to mass ratio:
$$ P/V = I = (m_i dN_i/dt) q_i/m_i v$$
where the term in () is the total mass exhausted per second. Regrouping to highlight momentum:
$$ P/V = q_i/m_i (dm/dt) v$$
$$ P/V = q_i/m_i dp/dt $$
$dp/dt$ gives the thrust, so finally:
$$dp/dt = P/V (m_i/q_i) $$
More power and higher mass ions lead to more thrust; more specifically a higher mass/charge ratio is better.
