Inky pinky ponky game / Josephus problem
Sorry, but I don't think this code is very good. 

In create_c_list, the first node is allocated twice, leaking memory, and there
are no checks for allocation failure.  Parameters n_k and n_w are badly
named and the latter is not used.  The loop should be a for not while.
new_p could just as well be n, for 'node'.  Also, it is not obvious that
the 'c' in the name means 'circular' and you should add spaces around
operators ('=' etc).

In elimination, the function name should be a verb.  The outer while(1) loop is often preferred as for (;;) , although that is a matter of personal preference. Indefinite loops are usually best avoided anyway.
Inside this loop is a for loop, which should define iter in the loop and call it i.  Parameter cntr means nothing - it could be
'counter' of 'center'.  Give it a better name.  It derives from a value
entered by the user called num which gives away no secrets, and the prompt
was "How many no. of words?" - so num and cntr are the number of 'words',
but the list created by create_c_list makes no mention of 'words'.  It is as
if you don't want the reader to understand what it does.  The for loop locates the node according to the cntr, which takes a long time if cntr is zero, as the loops starts from 1.  Then you do another
loop, this time a while in which you find the list entry before the one
(called it) that you located in the for loop.  You could have combined the two loops.  You then delete it using an unnecessary temporary
variable.
And so the outer loop continues until there is only one entry in the list - that condition could have been the condition for the outer while loop.

In printList (which uses camelCase while create_c_list doesn't)  you don't
modify the list, so it should be passed const.  Parameter list_tail is
inaccurate - a circular list has no tail.  Call it just list. 

Also, putting main last avoids the need for prototypes, all functions except
main should be static and the names of struct ll_core and c_ll would
be better as struct node and value.