Ich wollte, ich könnte mit.
According to the translation I could find on tatoeba.org, it means
I wanted to do it, but could not.
but I can't work out what the mit is for.
The translation from tatoeba is wrong or at least not 100% what it says.
The sentence could have several meanings, the most obvious one would be:
I wish I could come along.
where mit would be short for mitkommen/mitgehen/mitfahren.
Another, less obvious meaning could be:
I wish I could [do sth] with [using sth].
where the parts in brackets have to be clear from context (as they are left out) and mit=with. Typically you would say this in reply to some kind of "request" which also contains the word "mit".
An example for this second meaning: You are making snowballs with bare hands because they turn out better than when you do them with gloves on. Your friend is telling you: "Your hands are getting cold. Try with gloves!" To which you reply: "Ich wollte, ich könnte [Schneebälle] mit [Handschuhen formen]". Parts in brackets can be left out and I marked the "mit/with" to point out that it appears in the request and the reply.
There are verbs of movement:
You can use all this verbs to describe a movement into a certain direction, or from one place to another:
Walter geht ins Haus. Walter goes into the house.
Lola rennt nach Norden. Lola runs to the north.
But sometimes you describe a movement with a verb that doesn't really describe a movement, but a desire ...
Walter will ins Haus. Walter wants into the house.
Lola möchte nach Norden. Lola wants to the north.
... or an ability:
Walter darf ins Haus. (Walter is allowed to enter the house.) (sorry, there is no 1:1 translation)
Lola kann nach Norden. (Lola is able to go to the north.)
Those usages have developed from sentences like this:
Walter will ins Haus gehen. Walter wants to go into the house.
Lola möchte nach Norden fahren. Lola wants to drive to the north.
Walter darf ins Haus laufen. Walter is allowed to run into the house.
Lola kann nach Norden segeln. Lola is able to sail to the north.
But in this cases the kinds of movement often is not so important, or it is well known from the context. So in most cases you just want to express, that you can move, or that you want move, without spending much time on describing the exact kind of movement.
So, in German, people just left out that verb of movement, and what you've got now are sentences, that are considered to be grammatical correct, but only have a verb of desire or ability that now describes a generic movement.
»Mitkönnen« is a very special verb, that from it's origin is a verb of ability (können = to can; to be able to), but together with its separable prefix mit can only be used in the manner described above as a verb of generic movement.
How did this happen?
In German you can add the prefix »mit« to a verb of movement to express, that you are moving together with someone or something:
Ich werde mit Michael mitfahren. I with drive together with Michael. (Literally something like: »I will co-drive with Michael.« But that's not English grammar.)
or in present tense (learn, that »mitfahren« is a separable verb):
Ich fahre mit Michael mit. I drive together with Michael. (Literal: »I will co-drive with Michael.«)
Note, that the bold and italics »mit« at the end of the sentence is the »co-« in the literal translation, while the »mit« in the middle of the sentence translates straight forward into »with« in English.
Ich fahre bei Michael mit. (Literal: »I co-drive at Michael.«)
Here we have the preposition »mit/with« replaces by »bei/by«. The meaning is almost the same, but now it is even more clear, that Michael is the driver and that he drives me. But the »mit« at the end didn't change! This is a part of the separable verb!
This mit-prefix can be added to all verbs of movement where it makes sense, that someone moves together with someone else or something else.
(present tense:) Die Schraube dreht sich mit der Achse mit. The screw rotates together with the axis.
(future tense:) Die Schraube wird sich mit der Achse mitdrehen. The screw will rotate together with the axis.
But when it comes to the case, that you want to replace the verb of movement with a verb of desire or ability to describe a generic movement, and when you still want to say, that you are moving together with someone/something, then you have to plug the prefix mit to this verb of desire or ability:
Doris darf mit uns nach Italien mit. (Doris is allowed to go/drive/fly together with us to Italy.)
Doris wird wahrscheinlich nicht mit uns nach Italien mitdürfen. (Doris probably will not be allowed to go/drive/fly together with us to Italy.)
Your example sentence combines:
And as extra bonus you also have a subjunctive (»ich wollte« instead of »ich will«)
So the full translation of
Ich wollte, ich könnte mit.
I wished, I could go with you/them.
I wished, I could [go|fly|drive|sail|dive|rob|...](select what fits best into the context) together(in the sense of accompanying) with (someone who is not inside this sentence, you have to find him/her in the context).