Upcast spells (cast with a 6th level slot) count as 6th level spells, but it's unclear whether this would apply to the Horrible Scar rule
In the PHB, p. 201, under Spell Slots:
When a spellcaster casts a spell using a slot that is of a higher level than the spell, the spell assumes the higher level for that casting. For instance, if Umara casts magic missile using one of her 2nd-level slots, that magic missile is 2nd level. Effectively, the spell expands to fill the slot it is put into.
If cure wounds (which is usually a 1st level spell) is cast with a 6th level spell slot, that casting of cure wounds is actually a 6th level spell. This could be considered satisfactory for the Horrible Scar clause about removing the injury with healing magic of 6th level of higher.
On the other hand, Horrible Scar specifically gives the examples of heal and regenerate, which are both spells of 6th level or higher. They didn't list an upcast cure wounds, so you could also argue that the rule is referring to the spell's "base level" rather than the level it would be if you upcast it. Under this interpretation, an upcast cure wounds wouldn't work.
Of course, you could argue that because RAW doesn't say that you can't use an upcast cure wounds, then an upcast cure wounds counts, but actually what this really means is that RAW is unclear and it's down to the DM to decide whether an upcast cure wounds counts, or whether it has to be heal or similar.
Personally, I'd say it would be alright to allow a 6th level casting of cure wounds to fix the problem, since a 6th level spell slot is being spent either way. If we're talking about clerics or druids1, then most likely they could both have access to and cast heal anyway if they wanted to (and if they had it prepared that day).
In fact, whether one has it prepared or not is probably the only reason this would ever matter. As the DM, if the caster who is willing to expend a 6th level spell has cure wounds (or similar) prepared but not heal (or similar), it's up to you to decide that there are time sensitive reasons why it might be interesting for cure wounds not to work (like an important meeting with a vain noble who would look down upon such scars, and they're about to enter the meeting - not tomorrow, but right now! Or something like that...)
If you aren't the DM, then it's up to your DM.
1 - It's most likely going to be a cleric or druid casting 6th level healing spells, but it could be a bard, or a multiclassed wizard/ranger or something, in which case, since they might not have access to heal or similar at all, such a ruling becomes significantly more important for reasons besides timing and which spells are prepared... If that's the party's only option, I'd still allow it, unless seeking an NPC cleric or something would be more interesting for the narrative.