He looks too frail to actually chase and catch any live insects.
He should have been on at least TWO MEALS OF LIVE HIGH QUALITY INSECTS (THREE IS BETTER) AT 4 WEEKS OLD, I suspect a lot of the issue may be malnutrition (insufficient dietary calcium and insect protein + too cold to digest any food he was given + no UVA and no UVB).
This is one sick little hatchling , but he can recover and start to thrive with THREE LIVE INSECT MEALS PER DAY (IF HE CAN CATCH BUGS - UNLIKELY AT THIS TIME) or by being FORCED FED THREE TIMES PER DAY using something like HERPABOOST, or CARNIVORE CRITICAL CARE ór BUGPIE made as a slurry and given to him by feeding syringe and crop/feeding needle , you''ll likely need to give him liquid calcium daily orally by syringe (something like CALCIVET).
You can likely get HERPABOOST (or it's equivalent from a good reptile vet) , see
http://www.vetnpetdirect.com.au/vetafarm_herpaboost
CALCIVET from a good petshop maybe (or the vet) , see
http://www.vetnpetdirect.com.au/vetafarm_calcivet_liquid_calcium_supplement ,
he needs this in order to get his calcium levels up as dusting his insects will be too hit and miss for him at the present, and so he can produce the needed VitD on exposure to HIGH INTENSITY UVB (AT LEAST 10% from the MVB, or compact (26W UVB200 is minimum for your tank size, placed about 6-8 inches from his basking spot (on the ground as he's too frail to climb safely
), or 10%-14% UVB T8 or T5 tube.
I think Oxbow Carnivore Critical Care is only available from vet see
http://www.oxbowanimalhealth.com/products/type/detail?object=1609
BugPie can be bought from good pet shops or by mailorder see
https://www.store.repashy.com/by-product-name-en/grub-pie-reptile-en/ , made from BSF maggots and so naturally high in calcium and protein.
Crop Feeding Needles are available from good pet shop or from your vet see
http://www.vetnpetdirect.com.au/crop_medication_needles_birds?size=348
Feeding syringes are available from good pet shops or your vet see
http://www.vetnpetdirect.com.au/reusable_feeding_syringe_birds?volume=95
I have attached an image of how to use crop needles to feed a lizard :
He only has a very small capacity stomache , so you will likely be giving him 1 - 4 ml / 100g of bodyweight, so if he only weighs 10g that meals 0.1 - 0.4ml ) of liquid/slurry/pureed food / food supplement per feed and will likely need to use 14Gauge or 12Gauge crop / feeding needle. The vet or his nursing staff can show you how to do this if you ask them to .
Probably a good idea to let the vet care for him for a week or so to get him started under intensive care.
It will take a while (several weeks) to fatten him upo and get him to the point where he is able to cope with catching and eating insects like roaches and crickets, I'd start him off on BSF maggots or small (1.5 inch) silkworms , these are soft and slow moving and very easy for little sick lizard to digest.
As already said by others
he needs to kept warm 24/7 >> daytime warm zone about 30oCelsius, basking spot about 40-42oCelsius, overnight warmer than 24oCelsius (I'd go 26oCelsius to help him digest his food and boost his immune system).
UVA greater than 30%
UVB greater than at least 10%
and I think I'd bump up his photoperiod (time the UV and basking lights are on each day) to 16 hours --> more time for him bask, stay quite warm, digest his food, soak up UVA and UVB.
The UVA will help enhance his activity levels and appetite.
The UVB is necessary for him to be able to metabolise his dietary calcium and produce VitD --> make strong dense bones, and grow more of himself.
I have no doubt in a month or two he'll be growing like a weed and thriving and eating you out of house and home , and your pride and joy and a lovely little beardie.
Just going to a lot of TSL and intensive care from you and a bit effort to get him past this.