This is about the usage of the definitive accusative case (-ı/-i/-u/-ü) rather than possessives.
The suffix is added to objects. Most of the objects (>80%) have it.
If the suffix is not added, this is most likely because the object is newly introduced.
Objects in possessives always have this suffix.
A simple example (without possessives):
Ben kedileri severim. (I like cats)
Kediler beni sever. (Cats like me)
"I" became "me". This is expressed with one of the four suffixes above.
"Cats" don't change in English, but in Turkish the same rule, adding
I added detailed description of when to use and when not to use the definitive accusative suffix.
Some examples without the accusative suffix, with questions they're good answers to:
Yerde para buldum (I found money on the ground) (What did you find?)
Bazen hikâyeler yazarım (I sometimes write stories) (What do you do to improve your writing skills?)
Ona mektup yazdım (I wrote him/her a letter) (What did you do for him/her?)
Çekilişe bilet aldım (I bought a ticket for the lottery) (What did you buy?)
Garaja üç kalem bıraktım (I left three pencils in (to) the garage) (How could you expect me to draw in the garage without any drawing equipment?)
Adding the suffix would require some changes:
Parayı yerde buldum (On the ground I found money) (Where did you find the money you lost?)
Genelde aklımdaki hikâyeleri defterime yazarım (Usually I write the stories in my head to my notebook) (How do you record your stories?)
Mektubu ona yazdım (I wrote the letter to him/her) (This is for whom?)
Bileti şanssız buldum. (I found the ticket to be unlucky) (Any luck with the lottery?)
O üç kalemi garajda bıraktım (I left the three pencils in the garage) (You had pencils in your had, but I can't find them in your drawers in which you put rulers?)
In Turkish, the word before the verb is stressed. Objects in the second group aren't introduced in those sentences, but those sentences add information to those objects that were already known. That was why those objects don't "deserve" to be before the verb. Without the suffix, they're newly introduced.
A note about the last example: "O" (3rd singular pronoun) is added added in the meaning of "those" as in "those X things" to numbered, plural, specific objects with the suffix.
Examples 1, 3 and 4 feature singular objects. An optional "bir" can be added without the suffix without any change in meaning:
Ona bir mektup yazdım.
Çekilişe bir bilet aldım.
1 is an exception. "Bir para" is awkward.
bir before an object with the suffix is incredibly rare and unneeded.
Make sure not to add the suffix to idioms like "fotoğraf çekmek". Those have their separate entries in https://sozluk.gov.tr "fotoğrafı çekmek" is awkward.
When a verb is the object, it might or might have the accusative definitive suffix, which I'm not covering in this answer..