Perhaps he's running with Tom.
这里没有任何语法错误，是正确的.
不用run，用running，be动词后面一般加动词的ing形式.
perhaps的解释
ad. 也许，可能
perhaps的例句：
Perhaps I am wrong， but I think she is near forty.
也许我错了，但是我想她快四十岁了。
Such a mistake would perhaps lead to disastrous consequences.
这样一种错误可能导致灾难性的后果
