That quote proves nothing, except the judge was unwilling to take financial hardship or special circumstances into account for your challenge. Which is appropriate, because the law as written does not allow for that. (Infractions like traffic violations are generally fixed penalties.)
If you had wanted to challenge the validity of the ticket and the judge said he didn't care about the facts, that would have been a different thing altogether.
A judge can absolutely reduce a fine at their discretion, the judge blatantly refusing to use his discretion and human judgement in favor of a black and white view of the law is why he could easily be replaced with a machine.
No, a judge cannot "absolutely" reduce a fine for an administrative violation or infraction at their discretion. It depends on the jurisdiction. With some rulebreaking, the punishment is all-or-nothing.
If you had wanted to challenge the validity of the ticket and the judge said he didn't care about the facts, that would have been a different thing altogether.