Translation:I see my English teacher every weekend.
Ownership can be implied sometimes in Chinese, for things that are close to you, like your house, your company and your family. "My father" could be 我的爸爸 but it could also just be 我爸爸. It's possible that this is the intended way to read this sentence.
But "an" or "the" should be accepted as well.
The 'my' here is necessary for clarity at least in English. For example, going to a doctor every weekend (could be different doctors, could be the same) would be significantly different than going to my doctor every weekend (implies a singular doctor that you're seeing).
You can apply the same logic here.
Can you help me? I'm looking for a friend of mine, she's an English teacher. - Be more specific. You are asking a barman working near a campus. I see an English teacher every weekend.
It seems teachers in the local school are mostly unsociable creeps. Except for one. I see the English teacher every weekend.
It's a small town with just two big marts, where you every time see a familiar face or two. I see my English teacher every weekend.
As a rule, when "周" means used as "a week", it is written as "週", otherwise it shares the same form with simplified one: "周". But many people mix these two characters together in Taiwan.
Besides, this sentence sounds strange to me since "見/见" sounds like you just meet him and have a casual talk, but not learning English. I am not sure if this sounds fine for people in mainland China.