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

For non-negative integers n, r, s, t, let Aₙ^ (r, s, t) =₊=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 \Aₙ^{ (r, s, t) \}₍ ₀ in the form: align* A₍₏^ (r, s, t) Aₙ^ (r, s, t) +p³B₏-₃A^ (r, s, t) ₙp⁴, 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 B₏-₃ is the (p-3) th Bernoulli numbers. This type of supercongruences for another two Ap\'ery-like numbers are also established.