The woefully underclued crossword

This puzzle is part of the Monthly Topic Challenge #4: Cross-*non*-words

Giving clues for everything makes things way too simple. In this crossword, four clues are not given and you must find a way to fill every cell of the grid using the clues provided, including finding suitable answers to the unclued entries.

1. Portrayed by Hugo Weaving, this film character hides his face behind a Guy Fawkes mask
2. Expression for "softly" in sheet music
3. Notation for a kiss at the end of a letter, for example
4. The symbol on the front of automobiles manufactured by Elon Musk's company
5. –
6. On treasure maps, the spot where one should dig
7. Mechanical work, in physics
8. In chess notation, the piece that can jump over other pieces
9. Number whose square is −1
10. –
11. On pencils, "fine"
12. –
13. –

1. Portrayed by Hugo Weaving, this film character hides his face behind a Guy Fawkes mask : V
2. Expression for "softly" in sheet music : P
3. Notation for a kiss at the end of a letter, for example : X
4. The symbol on the front of automobiles manufactured by Elon Musk's company : T
5. :
6. On treasure maps, the spot where one should dig : X
7. Mechanical work, in physics : W
8. In chess notation, the piece that can jump over other pieces : N
9. Number whose square is −1 : I
10. :
11. On pencils, "fine" : F
12. :
13. :

At this point, given the tag, it looks like we are looking for

4 more PENTOMINOES (polyominos of 5 cells). They have a standard naming convention that matches the letters we found: Pentomino names

So let's start filling the grid:

(8,2) can't be reached by 4T. It can't be reached by 8N, or 4T doesn't fit. So it is part of 9I; forcing it.
This forces 4T and 3X, which forces 2P, which forces 1V.
I don't know if there are standard colors, so I'll choose some at random to meet the challenge criteria of not using letters in the grid.

(4,3) can’t be reached by 6X, so it is part of 7W, which is forced.
This forces 8N, which forces 6X, which forces 5 to be a U so that it doesn’t overlap 10.
New clue: 5 – 180 degree turn : U

There are two ways to fit 11F and leave room for 10, which must then be a P. Either way results in the same answers for the remaining clues, so we can pick one. I believe this might be resolved by moving the number '12' down a row.
New clue: 10 – Number greater than 1 divisible by only 1 and itself : P

At this point, the only thing that fits for 12 and 13 are two more P’s… and I’m running out of clue ideas so how about:
12 – the answer to 10 : P
13 – the answer to 12 : P

Completed grid:

• Exactly right, nicely done!
– Jafe
Oct 25, 2022 at 14:32
• @Jafe Is there a reason 11 can't go to R8C5 instead of R6C5? Oct 25, 2022 at 15:57
• @user39583 Interesting; I didn't notice that. That does have some implications. I went off the 'standard' unflipped shapes, I believe. Wait, no, a P is flipped...
– Amoz
Oct 25, 2022 at 16:01
• @user39583 Yikes, you're right. I missed that as well.
– Jafe
Oct 25, 2022 at 20:55
• @user39583 Unless I missed something, if you flip 11, the only solution for 12 and 13 is still to use two copies of P and flip 12 as well. Oct 26, 2022 at 6:52