On certain additive problems with primes and almost-primes

1) We consider the Diophantine inequality
$$|p_{1}^{c}\,+\,p_{2}^{c}\,+\,p_{3}^{c}\,-\,N|\,<\,(\log N)^{-E} ,$$
where $1 < c < \tfrac{15}{14}$, $N$ is a sufficiently large real number and $E > 0$ is an arbitrarily large constant. We prove that it has a solution in primes $p_{1}$, $p_{2}$, $p_{3}$ such that each of the numbers $p_{1} + 2$, $p_{2} + 2$, $p_{3} + 2$ has at most $\left[\frac{\displaystyle 369}{\displaystyle 180-168c\mathstrut}\right]$ prime factors (here $[t]$ denotes the integer part of $t$).
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2) (Joint work with Zh. Pertov). We consider the Diophantine equation
$$[p^c]\,+\,[m^c]\,=\,N,$$
where $1 < c < \tfrac{29}{28}$ and $N$ is a sufficiently large integer. We prove that it has a solution $p$, $m$, where $p$ is a prime and $m$ is an almost prime with at most $\left[\frac{\displaystyle 52}{\displaystyle 29-28c\mathstrut} \right]+ 1$ prime factors.