A 10th level Drow warlock (for instance) has the following Drow Magic racial trait (PHB p. 24), in addition to its warlock spellcasting abilities:
You know the dancing lights cantrip. When you reach 3rd level, you can cast the faerie fire spell once per day. When you reach 5th level, you can also cast the darkness spell once per day. Charisma is your spellcasting ability for these spells.
Can the Drow warlock cast faerie fire (which is not on the warlock spell list) using a warlock spell slot?
Can it cast darkness (which is on the warlock spell list) using a warlock spell slot?
The Sage Advice Compendium says this about the Magic Initiate feat:
If you’re a spellcaster, can you pick your own class when you gain the Magic Initiate feat? Yes, the feat doesn’t say you can’t. For example, if you’re a wizard and gain the Magic Initiate feat, you can choose wizard and thereby learn two more wizard cantrips and another 1st-level wizard spell.
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.
But this specifically requires you to pick the feat for one of your classes to be able to cast the spell learnt using your class spell slots, and does not seem directly applicable.
A Warlock can use warlock Pact Magic spell slots to cast spells learnt from levels of another spellcasting class, per the multiclass spellcasting rules (PHB p. 164):
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.
Similarly, if the warlock has learnt the invocation Mask of Many Faces (PHB p. 112)...
You can cast disguise self at will, without expending a spell slot.
...Can the warlock cast disguise self using a spell slot, e.g. to decrease the chance it can be dispelled?