The book is giving you some screwy examples. "Let his game be played by him" is correct but weird. You would only say something like that to make very unusual emphasis.
Here are a couple things to know.
How to make the subordinate clause
To make the object of let into the subject of its own clause, you need to put it into the objective case and put the verb into the infinitive. That's why you say:
Let him play his game.
rather than:
Let he plays his game.
Similarly, you would say Let him be helped by us, not Let he is helped by us. However, this sentence is equally as weird as Let his game be played by him.
Two (or three) senses of let
I think what the book is trying to do is teach two different senses of let at the same time that it's teaching you a tricky form of passive voice. Two of the main senses of the word let mean: (1) allow/permit the clause to happen; (2) suggesting or agreeing that "we" do the clause.
A classic example of the allow/permit sense: Let me go! is what a person who is being held against their will says to their captor.
A classic example of suggesting that "we" do something: Let's go! or Let's get started! is what you say when you want to start doing what you and your listener were just talking about doing together. This sense nearly always has us contracted to 's. (It has to be us rather than we because the subject of the clause has to be in the objective case, as above.)
Those are the most common and simplest examples to remember in order to learn the sound of the language. However, those examples can't be converted to a passive form.
Here's a more-realistic example in both active and passive form:
Let Dr. Kildare see the patient.
Let the patient be seen by Dr. Kildare.
Possibly your book has confused the suggesting/agreeing sense with a third sense of let, expressing a wish. A classic example of using let to express a wish is: Let peace prevail on Earth. Here's a realistic version of what I think your book is trying to demonstrate:
Let us beat our swords into plowshares.
Let all our swords be beaten into plowshares.
or, passively again, without using let:
May all swords be beaten into plowshares.
I can see why your book might have confused these senses. They really are a big, muddy mess. They are all variations on the basic sense of allow/permit, stretched to mean different things by repeated usage. The wishing sense can often be understood as the allowing sense and the suggesting sense simultaneously, where the request/suggestion is addressed to a deity, like Oh, God, please let there be peace on Earth.
The continuous form is used to state or ask what was, is or will be happening at a given moment. So, if you call your friend and ask: What are you doing? she may answer: I'm playing cricket (for example, she is standing on the boundary with her phone in her pocket when you call).
If you tell your friend that you intend to call her around 7pm tomorrow, she may reply: Sorry, but I'll be playing cricket. In other words, around the time that you plan to phone, she will be on the cricket field and unable to take your call. (The simple future I will play cricket is not possible in this context.)
In the passive form, the object is made the subject or topic of the sentence. So in a context where sports are the subject of discussion, the passive is just about conceivable.
Which sports will be being played after school tomorrow?
Cricket will be being played by me.
Tennis will be being played by John.
Badminton will be being played by Mary.
But as Parrott in Grammar for English Teachers (p336) states:
Some people dislike putting two forms of be together (e.g. be being or
been being) ... They avoid standard passive constructions in the
future continuous or predent perfect continuous.
Perhaps you could ask your friend why he or she changed your perfectly usual statement into one that would be perceived by native speakers as extremely odd. It would also be useful to link to the BBC English page referred to in the question.
Best Answer
There is no connection in meaning between "is played" and "has played"
"Is played" is a passive voice construction. "He is played by me" is equivalent to "I play him".
In the sentence "I play him", the subject is "I", the object is "him".
In the sentence "He is played by me" The grammatical subject "He" means the object of the verb. The subject is put in the "by me" phrase, and can be omitted. There is no grammatical object. A passive verb can never have a grammatical object.
So "He is played cricket" is grammatically incorrect English. The word "cricket" is placed as a grammatical object, but a passive verb never has a grammatical object.
On the other hand "has played" is an active verb form, and can have a grammatical object. "He has played cricket" is correct, just like "He plays cricket".
"I play cricket with him" is active. "Cricket is played by me with him" is passive (but we would never talk that way in natural English)