Perhaps, but it may not be recommended. Klenow can get into trouble if left unsupervised. Usually it is just incubated 15 min, or so, at 20-25C, with dNTPs (35uM) and then stopped with EDTA and heat (10mM, 75C 20'). You might get away with Klenow end filling, chilling, and adding ligase, supplement with the necessary ATP, and incubating at 16C or 4C, but the reaction is usually stopped first because the klenow polymerase can start running backwards and chew up the ends if incubated too long or warm.
Hmmm, if you were going to do only a 10 or 15 minute room temp ligation perhaps you could get away with it. You'd need to stop the reaction after 15 min, or conceivably, with luck, you could pop it directly into competent bacteria. I'm not sure if the klenow polymerase would have any feelings about the ATP being present for the ligase. This would be a blunt ligation, usually requiring an hour or two, so it may not get very far in only 15 minutes, although it is only an easy recircularization, so you might get away with it. High concentration ligase should work better.
I'm forgetting that the Hindiii must be inactivated so it won't cut the newly
Better would be to, cut the DNA with H3 + XhoI, heat inactivate 65c 20', add dNTPs and klenow, incubate 15 min 25C, add 10mM EDTA, heat inactivate 75C 20', add 20mM MgCl2 to replace the Mg++ removed by the EDTA (I was forgetting that), add ATP and ligase, incuabte 2 hrs, transform.
Am I forgetting anything? I think that should work, as long as the ligase doesn't get finicky about all the stuff present (it can be finicky about contaminants). You might increase the volume at the ligation stage to encourage circularization (which is favored anyway).
How do you check the final plasmid produced to confirm it is not a dimer? ? Run some uncut next to the original plasmid uncut?
If you left out the HindIII, you'd turn the Xho site into a PvuI site. Then, there would be no need to inactivate after restriction digesting.
If I weren't already so long winded already, I might add that the purpose of the gel extraction is more to remove any traces of uncut, supercoiled plasmid. A quick spin column would be sufficient to purify. Gel purification prevents background sometimes, but I think often a thorough digest can be sufficient, with luck.