No, Pact Magic spell slots cannot be used beyond casting Warlock spells, and Multiclass spellcasting.
The Warlock Class Feature, Pact Magic says this (emphasis mine):
Your arcane research and the magic bestowed on you by your
patron have given you facility with spells. See Spells Rules for the
general rules of spellcasting and the Spells Listing for the warlock
The Warlock table shows how many spell slots you have to cast your warlock spells of 1st through 5th level. The table also shows
what the level of those slots is; all of your spell slots are the same
level. To cast one of your warlock spells of 1st level or higher, you
must expend a spell slot. You regain all expended spell slots when you
finish a short or long rest.
The Feat Aberrant Dragonmark feat does not say that this spell learned counts as a warlock spell.
What if I'm a Sorcerer? Or a Sor/lock?
It does however count as a Sorcerer spell, in case you wanted to try a similar build with using a Sorcerer instead. Also, if you do become a Sorcerer/Warlock Multiclass, then you have access to this juicy paragraph under the Multiclassing options, (again emphasis mine)
Your capacity for spellcasting depends partly on your combined levels in all your spellcasting classes and partly on your
individual levels in those classes. Once you have the Spellcasting
feature from more than one class, use the rules below. If you
multiclass but have the Spellcasting feature from only one class, you
follow the rules as described in that class.
Pact Magic. If you have both the Spellcasting class feature and the Pact Magic class feature from the warlock class, you can use the
spell slots you gain from the Pact Magic feature to cast spells you
know or have prepared from classes with the Spellcasting class
feature, and you can use the spell slots you gain from the
Spellcasting class feature to cast warlock spells you know.
And the Sorcerer Spellcasting feature has the line:
The Sorcerer table shows how many spell slots you have to cast your sorcerer spells of 1st level and higher.
So if you take levels as Sorcerer, you can cast the Aberrant Dragonmark because you learned a Sorcerer spell, and if you Multiclass, Pact Magic slots can be used to cast these Sorcerer spells.
Now, why can we treat the spell learned from the Sorcerer list as a Sorcerer spell?
We can look at the Sage Advice Compendium, which addresses the similar Feat, Magic Initiate.
If you have spell slots, can you use them to cast the 1st-level spell
you learn with the Magic Initiate feat? Yes, but only if the class you
pick for the feat is one of your classes. For example, if you pick
sorcerer and you are a sorcerer, the Spellcasting feature for that
class tells you that you can use your spell slots to cast the sorcerer
spells you know, so you can use your spell slots to cast the 1st-level
sorcerer spell you learn from Magic Initiate. Similarly, if you are a
wizard and pick that class for the feat, you learn a 1st-level wizard
spell, which you could add to your spellbook and subsequently prepare.