Here is a simple exercise for those learning Carnatic music at the beginner level.
Take a swaram as your base note. For example let's begin with the lower Shadjam / Sa.
Reach directly from the base note to every other note until you reach the upper Shadjam. With Sa as the base note, we would sing Sa-Ri, Sa-Ga, Sa-Ma, Sa-Pa, Sa-Dha, Sa-Ni and Sa-Ṡa.
Repeat the exercise, taking turns to place each note in the centre of the circle to serve as the base note. For instance, with Ga as the base note, we would sing Ga-Sa, Ga-Ri, Ga-Ma, Ga-Pa, Ga-Dha, Ga-Ni, and Ga-Ṡa.
Repeat the exercise everyday until you are can connect notes that are further away from each other comfortably and precisely.
Once you are comfortable with this exercise, expand the circle to include the mandhara sthaayi (lower octave) and the thaara sthaayi (higher octave) notes which you can access.
Please note that this exercise assumes a fair grasp of the swara sthaanams / note positions. If you cannot locate e.g. the Ga swaram independently and correctly, it becomes difficult to connect Ga-Pa or Ga-Ni correctly.
Have a question? Leave it in the comments below.
Stay safe and keep singing,
P.S. Take a look at my Instagram page for more resources.