Yep. Cold solder joints fer shure. Cut the excess leads off, leaving about a 1/4-inch protruding above the copper side of the board. This will reduce the effect of the leads absorbing heat from the soldering iron. Re-heat each joint and remove the solder with a solder sucker or de-soldering braid. Clean around each joint with isopropyl alcohol and scrub vigorously with a tooth brush or short-bristled "acid brush". Rinse with more alcohol and blow dry (or allow to air-dry for a few minutes).
Make sure your soldering iron has a spade tip that is well-tinned. I would recommend at least 40 watts. Do not use a "soldering gun". Re-solder each joint with 60/40 rosin-core solder. Hold the spade tip between the lead and the PCB and apply solder to the joint. Aim for enough solder to "wick" up the lead wire and spread uniformly over the copper on the PCB. If the solder "balls up" like it does in your photo the iron and/or the solder joint is not hot enough. The finished solder joint should be bright and smooth. If desired, you may trim off any excess lead length after soldering. Flux-cored solder has just enough flux to do the job, and this flux should be burned away during the soldering process. If not, you may clean the soldered joint again with isopropyl alcohol, but this is not generally necessary because the solder flux is non-conductive. Avoid the temptation to apply an external flux from a can. Most of these are both corrosive and unnecessary and not good for electronics soldering.