How do you say this in Japanese? If I wanted to say I’m relieved my broken/imperfect Japanese is understood/comprehended (would 僕の片言の英語がなんとか通じるのでほっとしています be correct I feel like I have my verb tenses messed up lately)

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僕の片言の日本語がなんとか通じるのでほっとしています→This is perfectly fine, but you would be referring to your rather general experience of communicating with people or someone specific but in an ongoing relationship. For example, if you want to say this to someone who just had a conversation with you and praised your Japanese, you could say: 僕の片言の日本語が何とか通じてほっとしました。In that case you may want to use the past tense instead because you are referring to the specific experience that's already happened. (You would still say "I'm relieved" in English using the present tense in that context though so that's slightly different I think.)