use `FrameLabel` since you set Frame to True. You were using `FrameLabel` And also using `PlotLabel`. Why mix the two? `FrameLabel` can do all of these. `FrameLabel` is automatically centered on Frame.

![Mathematica graphics](https://i.sstatic.net/VZVoA.png)

    Plot[Sin[t], {t, 0, 50},
     PlotStyle -> Red,
     Axes -> False,
     Frame -> True,
     PlotTheme -> "Classic",
     FrameStyle -> Directive[Black, 25],
     PlotLegends -> Placed[{"Exact."}, {{1, 0.56}}],
     FrameLabel -> {{"ϕ (0, t)", Automatic}, {"t", 
        Style[Framed["w = 1, k = 0", RoundingRadius -> 5], 20, Black, 
         Background -> Lighter[White]]}},
     ImageSize -> 500
     ]