Hi-Fi RUSH

You unlock Peppermint as a partner on Stage two Production, giving you a ranged gun ability using . After this point you will start seeing certain enemies with a blue over shield (barrier) - such as the BA-B00 just before the Production stage boss fight.

If you shoot and destroy these "barriers" with Peppermint it should slowly progress the achievement tracking. You need to do this 10 & 50 times for both achievements.

Note: Removing the BA-B00's barrier in the training room doesn't count. Must be done in live play during any of the stages.

You first get access to using Peppermint as partner along with approacing enemies with barriers in Chapter 2 "Track 2: Power Up" but in Chapter 3 "Track 3: Trial By (Volcanic) Fire" you come across a bunch more enemies with barriers that you can start grinding it, simply hold RT while locking on a target with a barrier and release.