It depends on how good the password is, and the size of the hash prefix.
Large prefix / bad password
If we assume this is a hash of an average Joe's password which contains say 30 bits of entropy ("mySuperSecretPassword123" almost certainly contains less entropy than this), and to be conservative we follow Kerckhoffs's principle and assume the attacker knows how the password was generated, then there are only 230 possible passwords. If the prefix leaked is 80 bits from a SHA-1 hash, then it is extremely likely that there will be only one password candidate that matches the hash prefix.
log2(password space) is smaller than the leaked prefix, you might as well consider the entire hash to have leaked.
Small prefix / good password
What if the prefix is small or the password is good(ish)? Say for example you have a password space of 250, and you've leaked a 40 bit prefix. An attacker can't just crack the password, since there will be around 210 passwords that match the hash, but this is still a problem. 250 is much too large to launch an online attack, even without rate limiting. But if an attacker can pre-filter their guesses to those that match the prefix in an offline attack, they would only need to try 210 in the online attack, which may be feasible.
log2(password space) - prefix size > 0 then the attacker likely won't be able to crack the exact password, but if it's small enough, they can generate a pool of password candidates for use in an online attack.
Very good password
Of course, if you randomly choose from a password space larger than 2100 (to be conservative) with uniform probability, then leaking a partial or full hash is irrelevant, as it's never going to be cracked anyway.