When I was writing the lesson on ability (can) there were lots of example sentences with difference determiners (a / the). Hopefully this answers any questions you may have.
First, we'll look at some verbs we'll be using.
Vocabulary
Make
Play
Speak
Sing
Make
I can make pizza.
In this sentence there is no determiner. It is used to say we have the general ability.
I can make a pizza.
We have used 'a' as the determiner. This sentence means the subject (I) can make one pizza.
I can make the pizza.
We now have 'the' as the determiner. This sentence means the subject (I) can make one specific pizza
Conversation
Let's look at this conversation between friends organising a party.
A: We need food for the party. Can anyone make pizza?
B: Yes, I could make a pizza for the party.
C: Oh, I wanted to make the pizza for the party.
A: Okay, C, you can make the pizza.
A asks about the general ability of B and C.
B says they can make one (non-specific) pizza for the party.
C says they are going to make one specific pizza for the party.
Let's change the food to 'cake'.
I can make cakes.
There is no determiner because we are talking about a general ability. There is an 's' because 'cake' is being used in the countable form.
I can make a cake.
Here we have the determiner 'a' because you are talking generally about one cake.
I can make the cake.
Here we have the determiner 'the' because you are talking specifically about one cake.
Play
We can play lots of things: games, instruments, sports and lots more.
With games and sports, you do not need a determiner.
I can play football.
I can play chess.
You are talking about your general ability. With these you would never add a determiner.
Instruments are harder. In British English we use the determiner 'the'. I believe in American English the determiner is no used.
I can play the piano.
I can play the guitar.
I can play the drums.
The reason (I think) is because you are playing one instrument and so the ability is to play 'the' one instrument.
Speak
I can speak French.
For languages we do not use a determiner because they are proper nouns.
Proper nouns are any nouns of which there is only one. Countries, places and names are proper nouns. These never have a determiner. For example:
I went to France.
I went to France with David.
I went to France with David because I can speak French.
Other examples with ability:
I can read Shakespeare.
I can drink Jack Daniels whiskey.
I can understand Morse Code.
Sing
With 'sing' we can simply say:
I can sing.
This is because it implies a general ability to sing all songs.
Conclusion
General ability = no determiner (unless it's an instrument) and use the plural form if countable
Ability for one general item = a
Ability for one specific item = the
Ability for with a proper noun = no determiner