Can you change the direction of a car by poking it once with your finger? Yes. How much will it change? Not a whole lot. What if you could brace against something (the ground) and push really hard on the car with your finger for an hour? Well, the direction may change far more than it did from your quick little poke earlier. It may not be much, but when you figure in the huge distances in space, a small deviation can lead up to a big change over hundreds of thousands of kilometers.
The underlying physical principles are pretty much the same. An explosion generally doesn't have much mass and is over rapidly. It is a short impulse of a force. When you're trying to change the trajectory of a big object, it's not going to do the trick. A *big* explosion will rip up the ship but may have enough force to nudge it a bit. It's a tradeoff between damage and how much of a nudge you want.
Small ships pushing against a big one will sit there looking like they are doing nothing for a lonnnngggg time. But, eventually, they can start to push and prod and move that big ship around. It's like watching a tugboat move a barge full of coal. It's a slow process to change the direction of a large mass, but it can be done.
Are you writing science fiction or space opera? I'd argue that if it's the latter, then the needs of the plot and the story outweigh any major concerns about the physical realities of these kinds of situations. If you want the ship to keel over a bit because it took a hit from a Blastron X12 ship-to-ship Proton Torpedo and it threw Our Hero against a console that cut them above the eyebrow so now they are slightly injured and must soldier on in the face of a superior foe all to save their One True Love, well damn the (space) torpedoes and full (space) steam ahead!
If you want Our Hero to appear to nobly sacrifice themselves in the third Act by hopping into their trusty starfighter (the ironically-named Starlady's Luck) and bash it into the Gorlon Megafreighter that is bearing down on Orphania VI, the Planet of Orphanages, in a last-ditch effort to change its course. Well, if the Gorlons had their engine blown out and were still headed in the direction of Orphania, a push on their bow from a much smaller starfighter may do the trick. But only just. And you would need to apply a long impulse burn on full thrusters once you snuck up to the bow of the Gorlon ship. A long burn that is certain to overload the ship's fusion core reactor. Why, only a madman would come up with a suicidal plan like this to save people who (in Act I) were so cruel to them. A selfless gesture like this would fail miserably unless you figure that the Gorlon ship has plenty of distance before it hits Orphania, is no longer able to resist course changes due to an engine failure and that the Starlady's Luck has enough structural integrity to resist being smooshed by its own engine as it tries to act as a bow thruster.
https://en.wikipedia.org/wiki/Manoeuvring_thruster
Regarding your tethering question, space is kind of weird so you could imagine that a small ship that was using a reaction mass thruster could sit there blowing reaction mass out its backside all day - so long as the ship's engines were rated for a long-duration burn. The fact that it is tethered would mean a couple wrinkles - the strength of the tethering line vs the thrust of the engines determines if the cable snaps right away or if it holds. The location where the harpoon hits and holds the ship makes a big difference too. The structure that got harpooned could tear away if you throw the engines up to maximum causing worse problems for the small ship. Unless it was designed as a tug, it's probably not going to have hardened structures where you can safely attach a big line and drag along another ship.
Have fun with your story!
