Sure, a word that has at least one kanji can begin with kana. Generally, they come from combinations with independent words that are usually written in kana, or they come from combinations with prefixes like お- or ご-, although there are other prefixes that can form words like these as well.
For example, か弱い is now a single lexical word, originally consisting of 弱い plus the prefix か-:
か弱い
There's another possibility, too. Words are sometimes written in kana-kanji mixtures, particularly for technical reasons (a kanji is missing from a font) or because a given kanji is too rare or too difficult for the intended audience. For example, I once saw 一生懸命 written 一生けん命. And it's possible for the kanji which is replaced with kana to begin the word.
For example, 斡旋 might be written あっ旋 for this reason. When I search the Balanced Corpus of Contemporary Written Japanese (BCCWJ), I see 53 results for 「あっ旋」 and 281 for 「斡旋」. My speculation is that this is because 斡 is not a common kanji, listed as 2891 out of the 3500 most common kanji in the frequency chart published by the Agency of Cultural Affairs.
Also, as Tsuyoshi Ito points out in a comment, 斡 is not included on the Jōyō kanji chart. As discussed here, in some contexts non-Jōyō kanji are avoided, so this may be another reason why 斡旋 might be written あっ旋.
