A compelling lead who looks a little like Lady Day and sounds a lot like her, even when she sings, and a story with all the pathos built into it makes for a generally compelling screen biography of jazz icon Billie Holiday.

Lackluster direction and a shambolic script won’t make anyone forget “Lady Sings the Blues” or the recent documentary “Billie.” But “The United States vs. Billie Holiday” takes an original tack, making the case for her as a Civil Rights martyr persecuted by racist, vindictive Federal drug enforcement. That’s both a draw and a major drawback of this latest mixed-bag movie from Lee Daniels (“Precious,” “The Butler”).

In basing the script on a non-fiction history of the War on Drugs, “Chasing the Scream,” Daniels and actress-turned-screenwriter Suzan-Lori Parks (“Girl 9,””Their Eyes Were Watching God”) avoid spending rights money on the many fine Holiday biographies or scholarly history of the anti-lynching ballad “Strange Fruit” that Lady Day made famous. Cheaping-out resulted in a seriously disorganized, disjointed jeremiad that makes an impression and makes its points almost in spite of itself.

Grammy nominee Andra Day is Billie, and comes close to perfecting the intonation and affectations of Holiday’s singular voice — that timing, the little hiccups she twisted into the lyrics. This is a Billie who rarely looks the heroin-addled wreck barely keeping it together long enough to be a legend that is common in Holiday portraits. Her Billie seems sober enough, has a temper, isn’t shy about reminding us of the emotional scars that keep her abusing drugs and bouncing from one abusive relationship to another.

Flashbacks reinforce that, sketches of a childhood of abuse and coercion into prostitution.

The framework of the story is that most exhausted device, an interview, this time with a flamboyantly gay fan (Fictional?) radio interviewer (Leslie Jordan).

Lady Day half-retraces her career in this interview, building this narrative not around the major abusers in her life, husbands (Erik LaRay Harvey) and managers (Tone Bell), but around the Federal narcotics agent (Trevante Rhodes of “Moonlight”) who dogged her footsteps for years. Jimmy Fletcher gets out of the Army at the end of World War II, goes to see Holiday in concert, tries to get close to her and eventually does.

She figures out he’s a Fed, but falls for him (in this story) as he in turn recognizes that his racist boss, Harry Anslinger (Garrett Hedlund) has a vendetta against her and that the entire Federal government is out to stop her from singing “Strange Fruit.”

That’s more truth than fiction, but not remotely the literal truth.

The film skips back and forth in time, tracing the postwar decade-and-a-half of Holiday’s life and career, using drugs and misusing her band (Tyler James Williams plays her most famous accompanist, Lester Young), having a fling with movie star Tallulah Bankhead (Natasha Lyonne) and falling in and out of an affair with the Fed who may have her back, or may be hellbent on putting her away.

“Be careful with this feeling. This love right here, baby, it won’t love you back. I promise.”

Love eludes her, something that impacts even her sexual appetites. “God Bless the Child,” one of her most famous numbers, wafts through the score in scene after scene, underlining how alone she is and how “got his (or her) own” is what matters.

The downward spiral is gradual, only truly evident near the end. But the persecution, the arrests and warnings about “that song,” are ongoing.

Day is quite good in a performance that de-emphasizes histrionics (mostly) and plays up Holiday’s self-preservation instincts. Her band is constantly griping about getting paid, as is her costumer (Miss Lawrence) and hairdresser/personal assistant (Gabourey Sidibe of “Precious”).

There’s some graphic drug abuse, some raw sex and a lot of music in the film, with “Strange Fruit” as the one number that Day performs, start to finish, in the third act. You can feel that this was what all involved wanted to be their organizing principle to the narrative, that song. But it isn’t.

Daniels is a seriously miss-or-hit filmmaker, and when he misses, he reaches for explicit sex and sexuality to save him. See “The Paperboy.” Or better yet, don’t. “Billie Holiday” treads lightly on Lady Day’s bisexuality and wallows in “likes it rough” nudity to make an obvious point about her inability to love.

That renders “United States vs. Billie Holiday” a movie you can appreciate on its own terms, especially if you know her story well enough to do the organizing Daniels and Parks don’t. But it’s a bio-pic that keeps its brilliant, sultry, complicated subject at arm’s length.

This was talked up as an Oscar contender, purely by virtue of its subject and star. It isn’t.

That doesn’t mean this fascinating performance and defensible if not wholly coherent message and point of view isn’t worth seeing. It is.

MPA Rating: unrated, violence, drug abuse, explicit sex, nudity, profanity, smoking

Cast:  Andra Day, Trevante Rhodes, Garrett Hedlund, Gabourey Sidibe, Tyler James Williams, Natasha Lyonne, Tone Bell, Leslie Jordan and Erik LaRay Harvey

Credits: Directed by Lee Daniels, script by Suzan-Lori Parks, based on a book by Johann Hari. A Hulu release.

Running time: 2:10