For each prime $p>3$ there are non twin primes $q,r$ with $p^3=2q+r$ Define $\mathbb P'=\{n\in\mathbb P|n-2,n+2\notin \mathbb P\}$.
Conjecture:

Given a
  prime $p>3$, then $\exists q,r\in\mathbb P':p^3=2q+r.$

Tested for the first 10000 primes. The solutions are not unique in general:
5 set-of-solutions cardinality . 1  ok
7 set-of-solutions cardinality . 1  ok
11 set-of-solutions cardinality . 10  ok
13 set-of-solutions cardinality . 23  ok
17 set-of-solutions cardinality . 41  ok
19 set-of-solutions cardinality . 38  ok
23 set-of-solutions cardinality . 82  ok
29 set-of-solutions cardinality . 166  ok
31 set-of-solutions cardinality . 171  ok
37 set-of-solutions cardinality . 253  ok
41 set-of-solutions cardinality . 391  ok
43 set-of-solutions cardinality . 411  ok
47 set-of-solutions cardinality . 604  ok

103 set-of-solutions cardinality . 4130  ok
107 set-of-solutions cardinality . 5755  ok

503 set-of-solutions cardinality . 382264  ok

Calculated using BigZ
 A: COMMENT.-I wanted to verify your proposition by an example. I realized that the problem itself is very hard and would like to ask you some questions.
1) $q$ and $r$ should be both positive?
2) There is no uniqueness, is not it?
3) I do not have numerical programs that allow me to go up to $10000$ but I tell you what I observed for $p=7$.
If $p^3=2q+r$ then I test the difference $p^3-2q$ for $q\le \frac{p^3-1}{2} $ and I see if the corresponding number fits.
I do that with $p=7$ then for the $39$ primes involved I have to remove $24$ twins so I test with $15$ primes (all this, of course, requires a particular calculation for each $p$ chosen at the beginning).
One has the set $S$ of primes concerned
$$ S=\{23,37,47,53,67,79,83,89,97,113,127,131,157,163,167\}$$
There are $4$ cases for equality $ 7^3 = 2q + r $ is fulfilled with $ q $ and $ r $ primes but $r$ is twin
$$ 7^3=2\cdot37+269=2\cdot97+149=2\cdot157+29=2\cdot163+17$$  and  only one case in which $r$ is not twin (in accordance with your guess)
$$7^3=2\cdot127+89$$
So appears the question whether there is uniqueness but it does not seem very plausible for large $p$.
My opinion is that your question is extremely difficult to answer, unless some easy thing to be presented to prove it is false (as is usually in number theory).
