There is no exact/correct definition of implementing MVP in Android
To answer your question, in my view the Presenter should not have any Android logic.
As such, the Adapter would be a "View" then i.e. Presenter provides it the data (via the Activity or Fragment) and it just deals with how to present this.
I'd do MVP as follow.
Model - POJO's, parsing, Storing (SQLlite) and retrieving data (http). Obviously I'd divide the POJO's, parsing and DB logic into sub folders - but this all falls into Model for me.
View - Activity, Fragment, Adapters - Activities & Fragment hold reference to a Presenter that gives them data to display. How this data/messages are displayed, look + feel etc. is dealt with in the View.
Presenter - The Middle man, provides the logic to inputs i.e. Button Clicks, retrieval of data, validation of inputs & then passes the result back to the View (Activity or Fragment)
Here's a great article on MVP
Here's a simplified diagram of MVP

Answer modified from this question (also answered by me)