I would definitely change his food. Harrington's is a very low quality kibble. Anything sold in supermarkets is unlikely to be a good food.
Composition: Maize, Turkey Meal (20%), Rice (15%), Meat Meal, Porridge Oats, Poultry Fat, Digest, Beet Pulp, Linseed, Minerals, Kelp (0.2%), Yeast (0.1%), Citrus Extract (0.04%), Yucca Extract (0.01%).
As you can see it is cereal based, (First ingredient is maize, but it also has rice and oats) and only 20% turkey, which is 'turkey meal' and 'meat meal' which means all the by-products of processing rather than actual meat. Cereals are just cheap fillers, not necessary for any dog, but especially not for Chihuahuas where every bite counts because they eat such small quantities.
Getting him onto a decent food with a quality source of protein will have an almost immediate impact on his skin/coat, although as Ashley has said, the hair loss may be genetic.
Have a look on dog food adviser, not many of the brands are available over here, but it is good for teaching you what to look for in a quality food. I would recommend a raw diet for a dog with skin issues, the next best thing would be Ziwi Peak, which you can get on Amazon. If you want to feed kibble then Acana or similar imported brands are better than any that are easily available here, James Wellbeloved is probably the best British made kibble but choose the grain-free.