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.
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
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.
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.
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.
With 'sing' we can simply say:
I can sing.
This is because it implies a general ability to sing all songs.
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