Emerging　IoT　applications　have　brought　up　new　coverage　problems　with　sink-based　mobile　sensors.　In　this　paper,　we　first　focus　on　the　MinSum　Sink-based　Line　Barrier　Coverage　(SLBC)　problem　of　covering　a　line　barrier　with　mobile　sensors　originated　at　sink　stations　distributed　on　the　plane.　The　objective　is　to　minimize　the　movement　sum　of　the　sensors　for　the　sake　of　energy　efficiency.　When　the　sinks　emit　sensors　with　non-uniform　radii,　we　prove　the　MinSum　SLBC　problem　is　$\mathcal{NP}$　-complete　via　reducing　from　the　Partition　problem　that　is　known　$\mathcal{NP}$　-　complete.　Then　for　the　MinSum　Sink-based　on-a-Line　Target　Coverage　(SLTC)　problem　of　covering　targets　on　a　line,　an　exact　algorithm　is　presented　based　on　grouping　the　targets　and　transforming　to　the　shortest　path　problem　in　the　auxiliary　graph　induced　by　the　vertices　corresponding　to　the　groups.　The　algorithm　runs　in　time　$O(n^{2})$　when　sinks　emit　sensors　of　uniform　sensing　radius,　and　in　time　$O(\vert　R\vert　^{2}n^{2})$　for　sensors　of　non-uniform　radii,　where　$n$　and　$\vert　R\vert$　are　respectively　the　number　of　targets　and　different　radii.　Eventually　for　SLBC,　we　propose　a　pseudo　additive　fully　polynomial-time　approximation　scheme　by　extending　the　algorithm　for　SLTC.　The　algorithm　runs　in　$O(k^{2}(\frac{L}{\epsilon})^{2})$　time　and　computes　a　coverage　with　total　movement　provably　bounded　by　$opt+\epsilon$　for　any　fixed　sufficiently　small　$\epsilon　＞　0$,　where　$opt,　k$　and　$L$　are　respectively　the　movement　of　an　optimum　solution,　the　number　of　sinks　and　the　length　of　the　barrier.　At　last,　experiments　are　carried　out　to　demonstrate　the　practical　performance　gain　of　our　algorithms.　©　2021　IEEE.