I’m playing a drow chaos sorcerer and want to take 1 level in either warlock or cleric. Both work with her character development. The problem I run into is utility vs damage. Currently she has 14 levels of sorcerer and her stats are STR 8, Dex 14, Con 14, INT 10, WIS 13, CHAR 20.

I have argued for both sides to myself and want to hear other’s opinions before I choose.

hexblade will give her a bonus action and any spell/cantrip i take uses CHAR. I could pick up two more damaging cantrips as I have a lot of utility ones. I would get two spells, and and one pact spell slot which I get back on a short rest) which I can always burn for a sorcerer point before a short rest, if I haven’t used it. 

Trickery cleric will give me access to guidance, light, and spare the dying. none of which require a good WIS mod and all of which are useful. And as for spells I can only prepare two a day but 1. I have access to all the first level spells, 2 clerics have spells, like shield of faith, that don’t require me to have a good WIS mod, and 3. That’s how many spells I would have with warlock. Also Trickery  clerics can give one person advantage on stealth for one hour infinitely, and there is only one person in the group who rolls with disadvantaged (my character is friends with them, and  just considers everyone else a meat shield or a DPS so she doesn’t have to use spell slots).

when it comes down to it my character really only uses cantrips as we are playing Dungeon of the Mad Mage, and we never know when we will get a long rest. I don’t use sorcerey points unless I want to bend fate for an attack roll or save, or twin cast a spell.

I'd probably go Hexblade. Especially as you mentioned that you don't get a lot of long rests... it gives you opportunities to stack sorcery points on short rests and gives access to Eldritch Blast for dealing solid ranged damage. Although if you're interested in more healing, maybe consider going for Celestial Warlock instead of Hexblade... it gives you Light and Sacred Flame by default in addition to your other known Cantrips, gives you access to a Cure Wounds that you'll be able to cast through CHA, and gives you a bonus action heal of up to two d6's that you can use as needed.

I'm going to agree with Transmorpher and add that it may even be worth going for a second level of Warlock to get invocations, combine Eldritch Blast with Agonising Blast and you're doing 3d10+15 damage. You'll also get a second 1st-level spell slot that recharges every short rest which you can burn for Sorcery Points - 2 points every short rest is pretty big. If you have Quicken Spell metamagic you can do 2x Eldritch Blasts in a turn for 2 points to deal 6d10+30, which will eventually become 8d10+40 when you reach 17th level. 

Plus, you get access to the Cleric Spell list. 

Definitely go Celestial Warlock, 2 level dip. You won't really be losing out on much - you'll still get 9th level spells, etc as well as the 18th level sorc capstone. 

Hexblade simply offers the most synergy with your Sorcerer, but really only if you also want to be weapon viable. If you do not care about using your Charisma for weapon attacks, Hexblade loses its appeal. Other patrons are still good, but the choice between Warlock & Cleric then boils down to whichever your gut instinct says to go with.

I also recommend warlock.

if your just going to dip 1 level i would say warlock.  If your willing to go 2 levels the channel divinity ability for trickery could be alot of fun to play with.

If only one level then Warlock may give you more flavor.  2 levels of either will significantly influence the character.

I would say that the suggestion for Celestial warlock is one you should consider, as it manages to fall neatly in between Warlock and Trickery cleric.

If we're talking about a single level, Celestial Warlock grants access to Cure Wounds on Charisma, as well as what amounts to two 'bonus' mini Healing Words via Healing Light.  Those are long rest, but they're still useful for picking someone up in an emergency.  You get four cantrips total, one more than cleric, though Sacred Flame is not really the greatest choice for damage.  You also get access to Armor of Agathys, which is stupid good and can either save you a lot of self-healing on your fragile sorcerer or deal nutty damage to stuff that rabbit-punches you instead of hitting you with trees.

Trickery, on the other hand, gives you access to Disguise Self and Charm Person as domain spells, one of which has a bad save because your wisdom is pants.  You gain access to the first-level Cleric list, half of which is mediocre because your Wisdom is pants.  You do gain medium armor proficiency from cleric (I think, anyways...) which meshes well with 14DX.  Three cleric cantrips and the anybody-else stealth thing, which mostly depends on how much you value your teammates as a drow arcanist.

Gonna throw my 2 cents into the Warlock column, for sure.  After having played a Cleric with mediocre Wisdom (I rolled really bad...) I can tell you you're not gonna want to play one with only 13 Wisdom.  

If you're wanting to get on with stabbing things, then definitely Hexblade.  But if you are looking to add more utility, Celestial is a fine alternative (if you can explain how a Celestial would take notice of a Drow, anyway).  I'd also recommend taking a peek at the Great Old one.  The spells it gives are actually quite interesting -- 3d6 psychic and eat reaction on a failed save, and Prone/Incapacitated on a failed save, concentration for up to a minute.  And just for being a GOO, you can telepathically speak to any target within 30 feet (they can't speak back though...).