Massless charged particles in an electric field According to this question, theoretically, there can be massless charged particles. 
What will happen if we put them in an electric field? 
How will they respond to the increase in momentum/energy? In case of photon the frequency of the associated electromagnetic wave increases in this case.
 A: In the linked question, the "fact" that in cosmology massless charged particles are thrown around in calculations of the primordial plasma seems to have escaped notice. Cosmology studies assume that before symmetry breaking gauge bosons and the fermions of the standard model table are massless.
At cosmological times, when quarks are massless, there will still be electric and magnetic fields built up by the charges of the quarks. This Zurich diploma thesis does some calculations.
Here is answered a related question, :

Synchrotron radiation from massless charge
Classical radiation power from an accelerated massive charge diverges in the zero-mass limit, while some authors suggest that strictly massless charge does not radiate at all. On the other hand, the regularized classical radiation reaction force, though looking odd, is non-zero and finite. To clarify this controversy, we consider radiation problem in massless scalar quantum electrodynamics in the external magnetic field. In this framework, synchrotron radiation is found to be non-zero, finite, and essentially quantum. Its spectral distribution is calculated using Schwinger's proper time technique for ab initio massless particle of zero spin. Provided $E^2$ is very much larger than $eH$, the maximum in the spectrum is shown to be at $(h/{2π})ω=Ε/3$, and the average photon energy is $4Ε/9$. The normalized spectrum is universal, depending neither on E nor on H. Quantum nature of radiation makes classical radiation reaction equation meaningless for massless charge. Classical theory is reliable only as providing the low-frequency part of the true quantum radiation spectrum.

This is in contradiction with a previous paper in Arxiv which claims that massless particles do not radiate.
It seems that massless charged quarks are  not a simple problem for cosmology, as this quite esoteric talk at CERN demonstrates. "1. A charged massless quark in a magnetic field - CERN Indico" .
The calculations are relevant for cosmological models and one expects that more results will be found in the future, as it seems not to be a trivial question for  energies before electroweak symmetry breaking.
I expect that an analogous study could be made for electric fields, but magnetic fields are more useful in the cosmological plasma states, as it is known from the studies of the sun that plasma can carry magnetic fields.
A: That is exactly what happens. Their energy and their momentum increase, although their spatial velocity would always be equal to c. You could observe this increase by scattering them with other particles to measure their energy.
A: 
According to this question: "Massless charged particles?", theoretically, there can be massless charged particles. 
What will happen if we put them in an electric field?

The electric field could theoretically alter the direction of these particles.

How will they respond to the increase in momentum/energy? In case of photon the frequency of the associated electromagnetic wave increases in this case.

Marek also says: "@Eelvex: it depends on your definition of theoretically impossible. But yeah, they are basically ruled out by experiment because a world with charged massless particles would be very different from ours. – Marek Apr 2 '11 at 9:34"
And to the other answer Eelvex says: "... massless particle must move at c. – Eelvex Feb 12 '14 at 13:12".
So it's not as though you will increase the speed. But, theoretically, a extremely minute increase would decrease the spacing between the peaks of the wavelength vector, increasing the frequency.
See Wikipedia's page Quantum Mechanics:
"Quantum mechanics differs from classical physics in that: energy, momentum and other quantities of a system may be restricted to discrete values (quantization), objects have characteristics of both particles and waves (wave-particle duality), and there are limits to the precision with which quantities can be known (uncertainty principle).".
Between the minute difference in speed you probably won't be able to create and the uncertainty principle your efforts won't be measurable. The energy required to increase the speed from a fraction less than $c$ (if they are going that slow) to $c$ would be immense. Slowing them would be easier, easier to do, easier to measure.

Thanks to anna v's answer I was promoted to investigate this question further, leading to this interesting discussion at ReseachGate: "How do I explain why a massless charged particle cannot exist?".
That lead to this paper: "Electrodynamics of massless charged particles" (21 Jan 2015), by Kurt Lechner (with 8 citations): "Abstract: We derive the classical dynamics of massless charged particles in a rigorous way from first principles. ...". - but it seems to lead to more open problems than an answer to your question.
The cited papers are more helpful, in particular: "Behaviour of Charged Spinning Massless Particles" (26 Dec 2017), by Ivan Morales, Bruno Neves, Zui Oporto, and Olivier Piguet:
"Abstract
We revisit the classical theory of a relativistic massless charged point particle with spin and interacting with an external electromagnetic field. In particular, we give a proper definition of its kinetic energy and its total energy, the latter being conserved when the external field is stationary. We also write the conservation laws for the linear and angular momenta. Finally, we find that the particle’s velocity may differ from $c$ as a result of the spin—electromagnetic field interaction, without jeopardizing Lorentz invariance."
Coincidentally, on pages 2 and 17, the paper mentions the electrons of graphene (as does the ResearchGate discussion) which references articles on condensed matter physics.
I'll leave this edit here if there's no further interest.
