The author first shows that every sufficiently large integer \(n\) can be written as the sum of a square and a \(P_3\) (an integer is called a \(P_r\), when it has at most \(r\) prime factors counted with multiplicity), and points out that in the latter representation, the \(P_3\) may be replaced by a \(P_2\) when \(n\) is not a square, by the work of \textit{H. Iwaniec} [Invent. Math. 47, 171--188 (1978; Zbl 0389.10031)]. When \(n\) is a square, the former statement is derived from Chen's celebrated theorem asserting that every sufficiently large even integer can be written as the sum of a prime and a \(P_2\). Next, the author mentions some relations between Goldbach's problem and representations of squares as sums of a square and a \(P_2\). These are based on the following observation; if \(2k=p+q\) with primes \(p\) and \(q\), then on putting \(m=k-p=q-k\), one finds that \(k^2-m^2= (k-m)(k+m)=pq\), whence \(k^2\) is the sum of a square and a \(P_2\). The converse argument is also valid in the situations where one may assure that \(k\pm m>1\). Further, several interesting numerical results on the number of representations of \(n\) as the sum of a square and a \(P_2\) are recorded for \(n\leq 10^5\). In the latter half of the paper, the author discusses some relations between Goldbach's problem and roots of the reciprocal polynomial \(f_{pq}(x)=(px^2-2nx+q)(qx^2-2nx+p)\). In particular it is proved that if \(n\) is a positive integer and \(p\), \(q\), \(r\), \(s\) are pairwise distinct odd primes with \(pq