First thing you'll want to do is inspect how deep the scratches are, but with some care. If you can feel the scratch, it will not be removable with any kind of poish and you'll need to repair it. If it's simply a deep scratch, you'll have to decide how much clear coat you are willing to remove to get it out. You must basically level the surrounding clear coat in order to remove it. If it is deep enough to require a lot of clear removal, then you could risk clear coat failure in the future or have to use some sanding steps to remove it. They probably aren't this bad, but just covering all the bases here.
So from here you'll have to use some kind of polish to remove the scratch. By hand, it is certainly going to be more difficult than by a quality machine, but at the very least you can round off the edges of the scratch, making it less visible. One product you might find easy to work with is Meguiars Scratch-X. Available pretty much anywhere and good results. Follow the directions on the label.
If that does not give you acceptable results, then you're going to have to step up to a more abrasive polish. At some point, you will not be able to use certain polishes by hand because you cannot generate enough heat to break them down. Think of polishes as grades of sand paper. Either your results will be ineffective or you will create small scratches that may or may not be removable by hand with a more mild polish.
If you do not have a quality machine, I would probably just stick with the mild polishes and try to round off the scratches to make them less visible. Locally available products are those like Scratch-X, Meguiars #9, 3M Swirl Mark Remover, etc. Anything stronger by hand doesn't make a whole lot of sense to me. Keep in mind that a lot of products, including those made by Meguiars, tend to have oil fillers that mask defects and once they wear off, you see the defect show up again. Like makeup for your car.