Thank you for your answer Daniel.
In fact I can play all major scales (and some minor ones too) quite easily (TU so not very fast though) but because I certainly learnt them the wrong way (repassing them in a mechanical manner) it does not seem to be very helpful neither when it comes to sight-reading nor when it comes to improvising (or try to !!!

).
So I am now reviewing them paying more attention to how they are constituted, naming each note as it is written on the music sheet (e.g. reading and saying "E" but playing Eb or E# because I am in a key where E only exists with a # or a b - a "mind view" that seems to work pretty well with me), naming its quality (tonic, dominant,...), etc.
=> I believe that should improve my sight-reading.
In parallel, I try to play notes belonging to a given scale in many different orders (like [123454322345654334567... and reverse] or [12131415161716151413121... and reverse] etc. where 1=Tonic, 2=super-tonic ,etc.) again keeping my mind aware of the quality of each note in the scale.
=> I believe that should help me "respect" the scale's key signature when improvising.
Finally, I try to play different rythm patterns/harmonisation/chors inversions with the left hand over the scale (or variation of it !).
=> I believe that should improve my knowledge of scales harmonisation (M,m,m,M,M,m,d,M), my hands independancy and also my capacity of improvising a right hand melody over a left hand rythm...
Impatient to hear comments on that approach !
Thank you,
Drooxy