As I am sure you know, flybacks are notorious for the high secondary currents. The only device that takes care of those spikes is the output capacitor. Howevr, since this capacitor is usually an electrolytic cap, it has ESR (equivalent series resistance). Basically that means your cap is really a cap in series with a resistance. The voltage ripple will be mostly dictated by that ESR:
Vripple=ESR*Isecpp
It should be clear that in order to reduce the voltage spikes at the output you need lower ESR caps. If that is difficult or impossible, then you need to parallel several caps. Their ESR will appear in parallel and hence reduced.
For your 4A supply, assuming you are running in discontinuous mode, the peak currents could be 8A or more. Since the peaks are 0.4V, it follows that the ESR appears to be 0.4V/8A=0.05 ohms. Normally, caps specifically designed for SMPS should have lower ESR than that, but not much lower. You can expect about 0.03 ohms. That is the maximum value at room temperature. Since the ESR increases at lower temperatures, make sure you have enough margin to meet the ripple spec at the minimum specified operating temperature. Typically, the ESR is 2-3 times greater at -10C and 3-5 times greater at -40C.
Normal P/S requirements are 1% ripple and noise. Your 400mVpp is therefore somewhat too high. The LC filter helps. But first of all, make sure you are measuring the noise correctly. Normally this is done with the scope probe ground lead removed and just a piece of wire wrapped around the probe tip ground sleeve, no longer than about 0.2" (5mm). The probe tip touches the output and this makeshift ground lead touches the ground right at the output capacitor. Unless you measure this way, you will always get confused about noise.
Also, test that the scope does not see any common mode noise. To do this, just short the probe tip to the ground lead and together to the P/S ground. Any noise you see now is just common mode noise. To remove it from the scope, use a ferrite ring and feed the scope probe through it several times (like you are winding a toroidal choke). That should help reduce the common mode noise to a great extent.
As for the regulation of your supply, it is less than 1%. Normal P/S requirements call for 1%, so you are about making it, although you do not have much margin. If this is a single-output supply it should be bettter than that, although I do not know what error amplifier you are using.
Anyway, when you measure the load regulation, any trace resistance can mislead you, so make sure you measure exactly across the output cap. By the way, the LC filter is ususally not in the regulation loop, so it will degrade the load regulation, since it has resistance, which is simply in series with the output.
To check if the poor regulation is due to the error amplifier or to trace resistance, make sure you measure the output voltage right at the junction points of the output voltage divider (that goes to the error amp). If the regulation appears better at those points then you know the problem is with trace resistance so you can try and lower it. If the regulation is still rather poor you need to see why the DC gain of the error amp is too low.