Jelly, 19 bytes

’ÆRḊμ_Æp=2Tịμ_2;μİS

Try it online!

I have a feeling that this is improvable, but I can't immediately see how.

Explanation to come.

Jelly, 19 bytes

’ÆRḊμ_Æp=2Tịμ_2;μİS

Try it online!

I have a feeling that this is improvable, but I can't immediately see how.

Explanation

’ÆRḊμ_Æp=2Tịμ_2;μİS
 ÆR Generate all primes from 2 to n inclusive
’ Subtract 1
 Ḋ Remove first element
’ÆRḊ Generate all primes from 3 to n-1 exclusive
 _Æp Subtract the previous prime (i.e. calculate the prime gap)
 =2 Compare to 2
 Tị Take elements of the input where the comparison is true
 _Æp=2Tị Filter a list of primes to the latter halves of prime pairs
 _2 Subtract 2
 ; Append
 _2; Append the list to the list with 2 subtracted from it
 İ Take reciprocals
 S Sum
 İS Take the sum of the reciprocals

The μ connect all these portions together pipeline-style, with each taking the output of the one before as its input.