Yes, because the damage is less than 1 foot across and caused by a single break or tear.
Before I go into more detail on your specific case, I want to observe that many people seem to read "a single break or tear" and think it means 'only one crack in one component' or 'one slash through one layer' (or even one fiber!), but that is an impractically narrow interpretation and leaves very little gameplay utility for this spell.
Saying "one break" is not the same as saying "one piece".
Mending "a single break or tear" "no longer than 1 foot in any dimension" could reasonably mean replacing a 10 inch hole ripped out of priceless painting, or an adobe wall, or a thatch roof, or the hull of a wooden boat. Those are not necessarily examples of only "one piece", but they are only "one break".
There is no explicit requirement that all pieces must be present.
In fact the description states that holes up to 1 foot are mended with "no trace of the former damage".
A cloak with a six inch patch of cloth bitten out of it can certainly be mended by this spell, by both RAW & RAI; therefore it seems unreasonable to rule that a map would be irreparable due to a missing piece.
Furthermore, a single break on a mirror may have more legs than a spider, and still be one break. A chip in a windshield isn't two or three breaks just because it is multifaceted. I'm not saying that Mending will repair a shattered vase all at once, but I am pointing out that "a single break" is not necessarily so finite as some seem to rule it.
To repair a small shattered mirror, multiple pieces would have to be allowed, but again, if those pieces are from one break, I do not see a problem with this. (Your GM May Vary.)
That said, I believe the original intent was to fully repair a small destroyed item, not to partially repair large ones, since objects have AC & HP, yet HP is not mentioned in this spell.
For the small, detailed item in question (the wooden backing of a mirror), there seems no doubt that it would be repaired in full, due to the damage being a single break less than 1 foot across. Level of detail and missing material are simply not specified as limiting factors.