  We consider the exceptional set in the binary Goldbach problem for sums of
two almost twin primes. Our main result is a power-saving bound for the
exceptional set in the problem of representing $m=p_1+p_2$ where $p_1+2$ has at
most $2$ prime divisors and $p_2+2$ has at most $3$ prime divisors. There are
three main ingredients in the proof: a new transference principle like approach
for sieves, a combination of the level of distribution estimates of
Bombieri--Friedlander--Iwaniec and Maynard with ideas of Drappeau to produce
power savings, and a generalisation of the circle method arguments of
Montgomery and Vaughan that incorporates sieve weights.
