Your weapon loadout is a bit problematic - it contains guns with very different projectile speeds and this makes targetting difficult. Also, you have only low-level weapons that cannot really put hurt on an enemy; the powerplant of the Barracuda can support bigger guns that will let you have more fun blasting your targets.
The Tanto guns are mediocre (if not worse) and while the Lavablades are basically okay - they are energy-efficienct and have a bonus against the graviton shields of Tekagi's forces - they are still a bit on the weak side for this late in the campaign.
I just flew missions 8 through 10 in a Barraduda with 4 x Lavablade Mk III and 2 x Magma Hammer Mk I, and the guns were adequate for the job. But there was a specific reason why I was using these low-level guns instead of the many decent or even excellent class 5/6 weapons that are available: I wanted to beat the campaign as fast as possible, without freelancing and without losing time by making detours to buy decent weaponry. I bought the class 5/6 Reapers when they became available a bit later; they were very satisfactory despite their somewhat mediocre stats and it was certainly fun to use them - I wish I had used Reapers instead of the bl**dy Death's Hand kit during mission 11 (JFTR: post-campaign savegame shows total time played just shy of 5:52 h
).
So I would recommend that you go back to an earlier savegame - Kyushu will do nicely. Then do two things: get stinking rich (around a quarter of a million but at the very least $100,000 credits; Juni will wait forever if need be) and get some decent guns for your Barracuda. Even the standard issue class 5 Scrambler or Sunfury guns will give you a much more satisfactory experience than your current loadout. If mission 9 gives you still problems after that, then buy the class 6 graviton shield at Kyoto before the mission. Advanced Thruster and Advanced Countermeasure dropper are recommended; Heavy Thruster and normal countermeasures are adequate for the job but the advanced thingies might give you a little edge.
Now specific tips.
Try to preserve a couple of nanos and batts during the initial skirmish with the patrol - you'll need them later. Afterwards, when you are cruising towards Tekagi's Arch, point your ship about halfway between the surface of the planetoid and the top of the arch. Then listen to the chatter; as soon as you hear something about incoming hostiles kill your engine. If you are lucky then the incoming enemies will not fire cruise disruptors at you. If you get a warning about incoming missiles then drop a couple of countermeasures (the disruptor missiles
can disrupt a kill-engine slide). If you get disrupted then you can still try a standard thruster + engine-kill getaway. The purpose of this is to buy you a little time during which you are alone with the four shield generators and the battleship nearby.
When you are at the Arch, target the generator nearest to the battleship first. This one is the hardest to take out because you do have little or no cover, but when you arrive your shield is fully charged and you'll have less problems dealing with this generator. When you work on the other gennies, try to put the shield sphere between you and the battleship; if the battleship doesn't see you it won't fire guns and missiles at you. Also, if you hear a warning about incoming missiles then do
not ignore it; if you are flying at normal or thruster speed then you can drop a flare but if not then you have to get moving. When you fly a pass at a gennie, watch your rear while you are firing guns and torps; if there are no enemy fighters firing at you and no incoming missiles then you can backpedal a bit with the 'X' key while the good opportunity lasts. This way you have to fly fewer passes.
Last but not least, once you have dispatched the gennies do
not dock with the Arch unless your shield has at least half charge, especially if your hull has taken damage that you cannot repair for want of nanos. There are high-powered missiles flying about that can take out an unshielded Barracuda in one or two hits. So, if your shields are low then fly a wavy evasive pattern until the shields have recharged a bit (do this also while you are working on the gennies - if your shields fail then break off and let them recharge). When you are ready to dock, select the Arch and fly close to the docking bay before issuing the Dock command. The docking bay is on the upper side of the Arch but it is not in the center.