t's always good to provide the context as "reading" can be used in several different ways.
* As a predicate which explains the action/event part of a sentence.
1 He reads a book - presente simple. (describes an habitual action)
2 He is reading a book - presente continuous (He is in the middle of the action).
3 He has been reading a book - present perfect continuous (the action continued from a past time point until now)
(In this usage, "having read something" is ungrammatical)
* In gerund phrases - the phrase acts as a noun representing the action (they can be the subject or object )
1. Reading poems gives me great pleasure.
2. Having read Das Kapital made me feel like I know something about politics.
#1 focuses on performing the action, while #2 on the effect after its completion.
That is, in #1, you have the pleasure WHILE you're reading; in #2, you feel more knowledgeable AFTER you finish reading.
* In participial phrases - the phrase makes a clause describing a parallel action, a cause, effect, etc. on another action.
1. I was sitting in my armchair reading a book. (parallel action/state)
2. Having read Das Kapital, I feel like I know something about politics. (the effect of an action after it has finished)
Similar idea as with the gerund case, but now they are acting as adverbial clauses modifying hte main clause's verb. Again, #1 focuses on what happens while the reading is going on, and #2 the effect remaining after the reading has finished.
The present perfect form [have + past participle] is a tense in its own right when used in the predicate. In gerund and participle clauses, it sets the clause in the past tense relative to the main clause.