Yes, this spell can ignore cover.
Here's the general rule for spell targeting, already quoted in the question (PHB, p. 204):
To target something, you must have a clear path to it, so it can't be
behind total cover.
And here's the rule that describes the effect of total cover (PHB, p. 196):
A target with total cover can't be targeted directly by an attack or spell, although some spells can reach such a target by including it in an area of effect.
(Emphases mine in both cases).
That is the entirety of the rules for the effects of total cover: total cover means you can't be the target of spells.
Here's the relevant section of the spell's description (Fizban's Treasury of Dragons, P.21):
You unleash a shimmering lance of psychic power from your forehead at a target that you can see within range. Alternatively, you can utter a creature's name. If the named target is within range, it becomes the spell's target even if you can't see it.
(Again, emphasis mine).
The spell provides a specific targeting rule ("A named creature within range") that beats the general rule ("A creature not behind total cover"). So, yes, Raulothim's Psychic Lance ignores cover if the caster knows and utters the target's name.
But the spell doesn't mention cover, just visibility.
The wording of the spell ("target that you can see ... even if you can't see it") does read as though the primary intent of the naming-a-creature option is to bypass concealment. But the stated effect is worded such that it clearly is also an exception to the total cover rules.
The "specific beats general" meta-rule of D&D (PHB, p. 7) doesn't require that more specific rules explicitly identify all of the general rules they create exceptions to. It's enough for the specific rules for (in this case) a spell to enumerate its effects, and they are understood to take precedence over any more general rules that might otherwise apply.
Doesn't total cover mean the creature can be a target, but the spell has no effect?
No, it specifically doesn't mean that. The total cover rule explicitly says that the effect of total cover is that a creature can't be targeted.
Here's how the rules word "targeted but no effect", from (for example) the Globe of Invulnerability spell:
...a spell can target creatures and objects within the barrier, but the spell has no effect on them.
That's not how the total cover rule is worded. Raulothim's Psychic Lance can target creatures in a Globe of Invulnerability, but it won't have any effect on them.