This peak is on the NMAs Group A list and requires a royalty fee to be paid...
Royalty for up to seven members: US $ 500
Royalty for each additional member up to 12 persons: US $ 100
Please note that Chukkung Ri is commonly spelled as 'Chukhung Ri' (or "Chhukung Ri"),
and Kala Pataar is commonly spelled as 'Kala Patthar'.
I was very surprised to see this peak listed at http://www.nepalmountaineering.org/nma_peaks.php
So officially a fee is required, but realistically nobody cares.
The Maoist plague is meanwhile gone. It is actually safer to take the (longer, but very pleasant) Jiri approach because this avoids the risky airplane landing in Lukla.
Chhukung Ri itself is sometimes listed as a trekking peak, but apparently only in error. Kyazo Ri 6186 meters is the actually peak that was supposed to be listed, but apparently someone goofed on the website (or so we were told). According to the National Park, you don't need a special permit for Chhukung Ri, in fact from their office and for a few rupees you can purchase a certificate for climbing it.