Using (10 087±44)×10^{6} J/ψ events collected with the BESIII detector at a center-of-mass energy of sqrt[s]=3.097 GeV, the antihyperon-nucleon annihilation processes Λ[over ¯]p→K^{+}π^{+}π^{-}+kπ^{0} (k=1, 2, 3) are studied at an incident Λ[over ¯] momentum of approximately 1.074 GeV/c. The reactions Λ[over ¯]p→K^{+}π^{+}π^{-}π^{0} and Λ[over ¯]p→K^{+}π^{+}π^{-}2π^{0} are observed for the first time, with corresponding cross sections σ_{Λ[over ¯]p→K^{+}π^{+}π^{-}π^{0}}=8.5_{-1.1}^{+1.2}(stat)±0.4(syst) mb and σ_{Λ[over ¯]p→K^{+}π^{+}π^{-}2π^{0}}=7.9_{-1.7}^{+1.9}±0.4 mb. No significant signal is found for Λ[over ¯]p→K^{+}π^{+}π^{-}3π^{0}, and an upper limit of 7.2 mb is set at a 90% confidence level. Evidence for the K^{*}(892)^{+} resonance is seen in the K^{+}π^{0} invariant mass spectrum M_{K^{+}π^{0}} for k=1, and the corresponding cross section for Λ[over ¯]p→K^{*}(892)^{+}π^{+}π^{-} is measured to be σ_{Λ[over ¯]p→K^{*}(892)^{+}π^{+}π^{-}}=12.5_{-3.4}^{+3.8}±1.2 mb. Owing to the limited statistics, possible interference effects are not considered. These findings offer crucial input to deepen our understanding of the antihyperon-nucleon interactions.