If I recall correctly, both engines now work well but make noises you don't care for. In the case of the first engine, it seemed likely to be a big-end knock. The second engine's noise you haven't described.
There are some characteristics of the noises you could identify, and perhaps these would suggest the cause.
One of the most common and least serious noises has the character of rattling tinware. It is usually caused by the cooling air cowl and/or the recoil starter hardware moving about or touching something.
Another common noise is a clack rather than a rattle, and happens precisely regularly in proportion to engine speed (at exactly half engine speed, perhaps from each of two different sources with different intensities). That is a side-valve tappet clearance noise, and is caused by those low-cost engines not having a "silencing curve" on the camshaft such as car engines use. After verifying that the clearance is actually correct, it can and should be ignored.
A less common noise is a tap rather like a big-end knock which occurs at engine speed and may be intense when the engine is cold-started, but fades as it warms up. That is piston slap, caused by excessive piston skirt clearance, and within reason it is of no consequence beyond being embarrassing in the presence of other people, who form negative opinions on our mechanical prowess when they hear it.
There are also two types of tap or rattle that occur at engine speed but do not change with warm-up. One of them can be of any intensity from a very light tap to a heavy thump. That is caused by excessive big end bearing clearance, and if it is of the heavy thump variety it should be cause for serious concern. The other tap or rattle is always quite light and is only of concern if we mistake it for a slack big-end. It is actually a gudgeon, or little-end, sound due to a poor fit between gudgeon pin and piston, or gudgeon pin and connecting rod. As long as its intensity is within reason, like piston slap it is again a social problem rather than a mechanical one.
Aside from these classic noises there can be rattles or taps due to relatively minor parts of the engine being loose or misaligned. My usual concern with those is whether they have recently arisen or have been there for years. If they have just begun they are more of a concern than if they have been there for years.
