covering the eyes

Covering the eyes and peeking.

There are at least two types of behaviour fitting this description, at least two reasons for doing this.

Type 1
This seems to be the type depicted in the above picture. It is a very conscious type. The child is pretending to not look at something. The child could just shut his eyes but shutting them and covering them with the hands is more emphatic and dramatic.
But he is looking, he is peeking through his fingers. He is lying really.
He is probably playing the game of hide and seek, but not playing properly. He is not obeying the rules of the game, possibly deliberately, but he is playing a game. He is quite an old child

Type 2
This type of the behaviour of covering the eyes is much more primitive, and I think we must look for it in a much younger child. We have no picture for this type as yet. The primitive logic and thinking involved in this type seems to be
I cannot see you, therefore you cannot see me.
In fact what we have here is
I cannot see anything, so I cannot see you
So you cannot see anything, and so in particular cannot see me
In general, any experience of the child will be thought to be an experience of other people also

Later the child outgrows this type of faulty thinking or logic, escapes from his early paleologic and his thinking becomes somewhat more like the Aristotelian ideal
The child who does this when the theme music for Dr Who begins, and also peeks through his fingers as his curiosity does battle with his fear, is acting quite primitively. (But if he runs and hides behind the settee, but peeks round the corner/edge of it he is acting much more rationally and realistically).
The thinking seems to have progressed along its twisted path. Now
I cannot see you, so you cannot see me
But I am peeking through my fingers so I can now see you but most of my eyes and face are still covered so you still can't see me (very well?)

Similar types of primitive/infantile thinking are seen in schizophrenia, which produces a regression in behaviour.
So we might have the/an explanation for the symptoms of
'thought broadcasting'

For more on the sort of stuff described under 'type 2', look at
Mad or Primitive Thinking
and the writings of
Jean Piaget
Sylvano Arieti