Mobile　edge　computing　(MEC)　paradigm　supports　cloud-like　computing　capabilities　at　the　edge　of　the　network　and　offers　low-latency　services.　Proxy　servers　of　MEC　with　mobility　and　limited　computing,　e.g.,　flying　unmanned　aerial　vehicles　(UAVs)　have　emerged　as　competitors　in　providing　services.　This　work　considers　a　task　offloading　problem　for　an　UAV-assisted　MEC　system　and　designs　an　integrated　cloud-edge　network　with　multiple　mobile　users　(MUs)　and　layered　UAVs　to　improve　MEC　with　a　network　of　UAVs.　In　our　system,　edge　UAVs　(EUAVs)　and　the　cloud　collaborate　to　provide　caching　and　computing　services　for　MUs.　We　consider　static　and　dynamic　applications　that　support　task　offloading.　Our　proposed　approach　minimizes　the　weighted　cost　of　latency　and　energy　consumption　by　jointly　optimizing　caching　and　offloading,　deployment　of　EUAVs,　and　allocation　of　computation　resources.　Simultaneously,　this　work　also　considers　UAVs＇　caching　and　computation　capacities　while　meeting　MUs＇　latency　and　energy　constraints.　Thus,　a　constrained　mixed　integer　nonlinear　program　for　a　layered　UAV-assisted　hybrid　cloud-edge　system　is　formulated.　To　solve　it,　this　work　designs　a　hybrid　metaheuristic　algorithm　named　adaptive　and　genetic　simulated　annealing　(SA)-based　particle　swarm　optimization　(AGSP).　Experimental　results　with　a　real-life　dataset　verify　that　the　AGSP＇s　system　energy　consumption　and　task　latency　are　reduced　by　at　least　7.4%　and　8.46%,　respectively,　compared　with　the　state-of-the-art　algorithms,　thus　proving　that　AGSP　greatly　enhances　the　energy　and　latency　of　the　system.