/diy/ - Do It Yourself

/diy/nobots,
is there a way to get past the plastic with connectors such as pic related?
i keep geting unstable readings with my multimeter, and i've pinned the problem down to the lead wires. i'd like to solder new wire to it, but is there any way to get past this plastic without fucking the connectors up?
what's that even called in english? molded connectors or what? (no it's not that, google images shows some special type of "molded connector")