There are two facets for this issues: one is conceptual and the other is technical.
The conceptual issue is that you need to define what you mean by built up area in your legend. A good starting point is to make the exercise to build a LCCS legend. It is indeed necessary to define what you want as a model before trying to map it. Note that in your case, a thematic map could be suboptimal but you could try to estimate the proportion of impervious surface inside each pixel. Furthermore, you should choose if your landscape is best represented with pixels of polygons (what would be your minimum mapping unit). The "scale" (or granularity) of your "heterogeneity" is therefore very important. Asking yourself if the heterogeneity is below or above the pixel should help you to take the good decision. This decision is particularly important if you want to perform a change detection, because you do not want to compare apples and pears.
From the technical point of view, manual delineation is great as long as you do not have a large area. Otherwise you could use image segmentation (if you are interested in a complex legend). There are segmentation tools available from third parties within QGIS (SAGA, OTB, GRASS). Note that Image segmentation will also aggregate pixels in the other land covers: this will reduce the salt and pepper effect, but you will loose some details. Alternatively, if your urban area is characterised by its texture due to heterogeneity, then you could use some pixel-based texture (e.g. a simple high pass filter in a 3*3 window).
As a remark, as mentioned by others, you should avoid Gaussian maximum likelihood if (i) you have a very large number of features or (i) the spectral signature of your classes id not normally distributed.