When you want to ligate sticky ends that are not compatible, you can fill up or bite off sticky ends with Klenow fragment (produced from recombinant truncated E. coli polA gene from which the 5'→3' exonuclease domain was removed), so that they become blunt. When you have a sticky 5’ overhang, it serves as template and you can fill the site up with the Klenow fragment and dNTP nucleotide mix. When you have a 3’ overhang, Klenow will digest off the overhang, which will also result in a blunt end (you still have to add the dNTP nucleotide mix because if the enzyme digests too far it will rapidly fill it up again). Digestion of 3' overhangs is much slower than filling up 5' overhangs, so if you have a 3’overhang, you should prepare yourself for a lower efficiency (which doesn’t mean that you will have problems).
Blunt ends can be ligated, I have made hundreds of constructs involving blunt end ligation. But nowadays PCR assembly is the preferred method when available restriction sites are not compatible or out of frame (in case of fusions within coding regions).
No, the Klenow fragment is a truncated form of DNA polymerase, it doesn't have the 5' to 3' exonuclease domain. You can use it for a PCR reaction, but it doesn't survive the denaturation step, so you would have to add new Klenow at every cyvle, during each annealing step. In the old days we did PCR like that...gosh the memories. We used to purify our own restriction enzymes in those days...but to come back to blunting sticky ends, you really need to use the Klenow fragment. The tube must have the word Klenow written on it...and don't forget the dNTP mix. 15 minutes at room temperature is all it takes (that's an excess time to make it foolproof)
To add to the conversation here, make sure you are not confusing T4 DNA polymerase and T4 DNA ligase. T4 DNA polymerase will fill in your ends by either 3' overhang removal or 5' overhang fill in, in contrast T4 DNA ligase with stick them together once you have blunt ends, but if you have incompatable ends and add T4 DNA ligase nothing will happen. The most important first step is to determine the kinds of ends that you have and decide if you want to remove overhangs or fill them in, if you are not sure check out this basic explination of blunting