Is it possible for a weak entity, already related to a strong entity, to get related to another weak entity in an ER diagram?
Depending on how much you briefed your problem statement, it can be stated that:

Trainer is a strong entity (it has an ID)

Neither Client nor Membership Plan seem to be weak or strong entities.
They both could be considered strong entities provided that a primary key is defined on them*; e.g. Client.(Name, Phone) and Membership Plan.(&quot;Beginning Date&quot;, Duration)


In your Chen diagram, Meets entity should have both ends total participation, but there's an arrow between Trainer and Meets?
Other than that, it seems to be alright.

* A strong entity is defined as having a key that does not depend on any other key; a weak entity as as having a key that depends on a foreign key. E.g. Order has a primary key order_id which is independent, and that makes it a strong entity, whilst Order_Detail has a primary key order_detail_id and unique key order_id, line_no, which makes it a weak entity since order_detail_id does not provide a meaningful independent ID, and the unique key depends on Order table. See weak entity entry in Wikipedia.
They're independent from other entities, so they are candidates for being considered strong entities provided that they have a primary key, which is not known if they have and is not clear if they could in the definition.