Supercongruences involving Ap\'ery-like numbers and Bernoulli numbers

For non-negative integers $n,r,s,t$, let Aₙ(r,s,t)=∑ₖ₌₀ⁿn kʳn+k kˢ2k nᵗ, which includes six Ap\'ery-like numbers as special cases. We establish one step deep congruences of the Gauss congruences for ₙ(r,s,t)≥ 0 in the form: {align*} Aₙₚ(r,s,t)≡ A_n(r,s,t)+p^3Bₚ₋₃A(r,s,t)_n{p^4}, {align*} for all primes p≥ 5 and all positive integers $n,r$ with r≥ 2, where A(r,s,t)ₙ is independent of p and Bₚ₋₃ is the $(p-3)$th Bernoulli numbers. This type of supercongruences for another two Ap\'ery-like numbers are also established.