A two-stage model for achieving X-ray amplifying action on photoionized K-transitions pumped by the soft X-ray broadband flux from laser-produced plasma is treated. A simple pumping criterion is considered by the analysis of the positive gain-length product for the two-stage model of innershell transitions (0) → (3) and (2) → (1). The incident photon flux integrated over the effective lifetime T21 of an innershell transition (2) → (1) is related to the photoionization coefficient μ03(ph), stimulated emission cross section σ21(stim), and the effective gainlength L(eff) ≃ 0.63/μ03(abs), where μ03(abs) is the absorption coefficient of a pump soft X-ray radiation. Numerical estimates for the feasibility of K-shell X-ray amplification are given for C, F, Na, Ti, and Sn. For the example of an exponential X-ray carrier pulse (XCP) propagating through a resonant medium, an amplifying effect depending on the ratio of the input pulse temporal bandwidth τband to the effective lifetime T21 is investigated. It is shown that for an XCP with a finite bandwidth τband the logarithm of an energy gain, ln[U(z)], falls below a linear relation with z and thus, from the viewpoint of maximum amplifying effect bandwidths τband with τband/T21 > 1 are preferable. The requirements for achieving X-ray K-shell photoionization amplification for C are discussed in more detail. PACS numbers: 32.80.Hd, 42.50.Md, 42.55.Vc, 42.65.Re, 52.25.Nr.